Around the Brunn-Minkowski inequality - UniFIweb.math.unifi.it/users/colesant/ricerca/Magdeburgo e...
Transcript of Around the Brunn-Minkowski inequality - UniFIweb.math.unifi.it/users/colesant/ricerca/Magdeburgo e...
![Page 1: Around the Brunn-Minkowski inequality - UniFIweb.math.unifi.it/users/colesant/ricerca/Magdeburgo e Berlino 2015.pdf · Around the Brunn-Minkowski inequality Andrea Colesanti Technische](https://reader031.fdocuments.in/reader031/viewer/2022030820/5b33fad37f8b9a6b548ba38c/html5/thumbnails/1.jpg)
Around the Brunn-Minkowski inequality
Andrea Colesanti
Technische Universitat Berlin - Institut fur Mathematik
January 28, 2015
![Page 2: Around the Brunn-Minkowski inequality - UniFIweb.math.unifi.it/users/colesant/ricerca/Magdeburgo e Berlino 2015.pdf · Around the Brunn-Minkowski inequality Andrea Colesanti Technische](https://reader031.fdocuments.in/reader031/viewer/2022030820/5b33fad37f8b9a6b548ba38c/html5/thumbnails/2.jpg)
Summary
I The Brunn-Minkowski inequality
I The isoperimetric inequality
I Infinitesimal form of Brunn-Minkowski inequality
I Inequalities of Brunn-Minkowski type
![Page 3: Around the Brunn-Minkowski inequality - UniFIweb.math.unifi.it/users/colesant/ricerca/Magdeburgo e Berlino 2015.pdf · Around the Brunn-Minkowski inequality Andrea Colesanti Technische](https://reader031.fdocuments.in/reader031/viewer/2022030820/5b33fad37f8b9a6b548ba38c/html5/thumbnails/3.jpg)
Summary
I The Brunn-Minkowski inequality
I The isoperimetric inequality
I Infinitesimal form of Brunn-Minkowski inequality
I Inequalities of Brunn-Minkowski type
![Page 4: Around the Brunn-Minkowski inequality - UniFIweb.math.unifi.it/users/colesant/ricerca/Magdeburgo e Berlino 2015.pdf · Around the Brunn-Minkowski inequality Andrea Colesanti Technische](https://reader031.fdocuments.in/reader031/viewer/2022030820/5b33fad37f8b9a6b548ba38c/html5/thumbnails/4.jpg)
Summary
I The Brunn-Minkowski inequality
I The isoperimetric inequality
I Infinitesimal form of Brunn-Minkowski inequality
I Inequalities of Brunn-Minkowski type
![Page 5: Around the Brunn-Minkowski inequality - UniFIweb.math.unifi.it/users/colesant/ricerca/Magdeburgo e Berlino 2015.pdf · Around the Brunn-Minkowski inequality Andrea Colesanti Technische](https://reader031.fdocuments.in/reader031/viewer/2022030820/5b33fad37f8b9a6b548ba38c/html5/thumbnails/5.jpg)
Summary
I The Brunn-Minkowski inequality
I The isoperimetric inequality
I Infinitesimal form of Brunn-Minkowski inequality
I Inequalities of Brunn-Minkowski type
![Page 6: Around the Brunn-Minkowski inequality - UniFIweb.math.unifi.it/users/colesant/ricerca/Magdeburgo e Berlino 2015.pdf · Around the Brunn-Minkowski inequality Andrea Colesanti Technische](https://reader031.fdocuments.in/reader031/viewer/2022030820/5b33fad37f8b9a6b548ba38c/html5/thumbnails/6.jpg)
Summary
I The Brunn-Minkowski inequality
I The isoperimetric inequality
I Infinitesimal form of Brunn-Minkowski inequality
I Inequalities of Brunn-Minkowski type
![Page 7: Around the Brunn-Minkowski inequality - UniFIweb.math.unifi.it/users/colesant/ricerca/Magdeburgo e Berlino 2015.pdf · Around the Brunn-Minkowski inequality Andrea Colesanti Technische](https://reader031.fdocuments.in/reader031/viewer/2022030820/5b33fad37f8b9a6b548ba38c/html5/thumbnails/7.jpg)
The Brunn-Minkowski inequality
Thm. A,B ⊂ Rn, compact; λ ∈ [0, 1]; then
Vn((1− λ)A + λB)1/n ≥ (1− λ)Vn(A)1/n + λVn(B)1/n. (BM)
I Vn = volume (Lebesgue measure);
I
(1− λ)A + λB = {(1− λ)a + λb : a ∈ A, b ∈ B}.
Equivalently: The functional V1/nn is concave in the class of
compact sets of Rn, equipped with the vector addition.
An excellent survey (much better than this talk):R. Gardner, The Brunn-Minkowski inequality, Bull. A.M.S., 2002.
![Page 8: Around the Brunn-Minkowski inequality - UniFIweb.math.unifi.it/users/colesant/ricerca/Magdeburgo e Berlino 2015.pdf · Around the Brunn-Minkowski inequality Andrea Colesanti Technische](https://reader031.fdocuments.in/reader031/viewer/2022030820/5b33fad37f8b9a6b548ba38c/html5/thumbnails/8.jpg)
The Brunn-Minkowski inequality
Thm. A,B ⊂ Rn, compact; λ ∈ [0, 1]; then
Vn((1− λ)A + λB)1/n ≥ (1− λ)Vn(A)1/n + λVn(B)1/n. (BM)
I Vn = volume (Lebesgue measure);
I
(1− λ)A + λB = {(1− λ)a + λb : a ∈ A, b ∈ B}.
Equivalently: The functional V1/nn is concave in the class of
compact sets of Rn, equipped with the vector addition.
An excellent survey (much better than this talk):R. Gardner, The Brunn-Minkowski inequality, Bull. A.M.S., 2002.
![Page 9: Around the Brunn-Minkowski inequality - UniFIweb.math.unifi.it/users/colesant/ricerca/Magdeburgo e Berlino 2015.pdf · Around the Brunn-Minkowski inequality Andrea Colesanti Technische](https://reader031.fdocuments.in/reader031/viewer/2022030820/5b33fad37f8b9a6b548ba38c/html5/thumbnails/9.jpg)
The Brunn-Minkowski inequality
Thm. A,B ⊂ Rn, compact; λ ∈ [0, 1]; then
Vn((1− λ)A + λB)1/n ≥ (1− λ)Vn(A)1/n + λVn(B)1/n. (BM)
I Vn = volume (Lebesgue measure);
I
(1− λ)A + λB = {(1− λ)a + λb : a ∈ A, b ∈ B}.
Equivalently: The functional V1/nn is concave in the class of
compact sets of Rn, equipped with the vector addition.
An excellent survey (much better than this talk):R. Gardner, The Brunn-Minkowski inequality, Bull. A.M.S., 2002.
![Page 10: Around the Brunn-Minkowski inequality - UniFIweb.math.unifi.it/users/colesant/ricerca/Magdeburgo e Berlino 2015.pdf · Around the Brunn-Minkowski inequality Andrea Colesanti Technische](https://reader031.fdocuments.in/reader031/viewer/2022030820/5b33fad37f8b9a6b548ba38c/html5/thumbnails/10.jpg)
(BM) privileges convex sets
In the inequality
Vn((1− λ)A + λB)1/n ≥ (1− λ)Vn(A)1/n + λVn(B)1/n,
equality holds iff A is convex and B is a homothetic copy of A (upto subsets of volume zero).
Why? If you plug A = B in (BM) in general you don’t get anequality, because
(1− λ)A + λA 6= A ((1− λ)A + λA ⊃ A).
But if A is convex
(1− λ)A + λA = A ∀λ ∈ [0, 1].
![Page 11: Around the Brunn-Minkowski inequality - UniFIweb.math.unifi.it/users/colesant/ricerca/Magdeburgo e Berlino 2015.pdf · Around the Brunn-Minkowski inequality Andrea Colesanti Technische](https://reader031.fdocuments.in/reader031/viewer/2022030820/5b33fad37f8b9a6b548ba38c/html5/thumbnails/11.jpg)
(BM) privileges convex sets
In the inequality
Vn((1− λ)A + λB)1/n ≥ (1− λ)Vn(A)1/n + λVn(B)1/n,
equality holds iff A is convex and B is a homothetic copy of A (upto subsets of volume zero).
Why? If you plug A = B in (BM) in general you don’t get anequality, because
(1− λ)A + λA 6= A ((1− λ)A + λA ⊃ A).
But if A is convex
(1− λ)A + λA = A ∀λ ∈ [0, 1].
![Page 12: Around the Brunn-Minkowski inequality - UniFIweb.math.unifi.it/users/colesant/ricerca/Magdeburgo e Berlino 2015.pdf · Around the Brunn-Minkowski inequality Andrea Colesanti Technische](https://reader031.fdocuments.in/reader031/viewer/2022030820/5b33fad37f8b9a6b548ba38c/html5/thumbnails/12.jpg)
(BM) privileges convex sets
In the inequality
Vn((1− λ)A + λB)1/n ≥ (1− λ)Vn(A)1/n + λVn(B)1/n,
equality holds iff A is convex and B is a homothetic copy of A (upto subsets of volume zero).
Why? If you plug A = B in (BM) in general you don’t get anequality, because
(1− λ)A + λA 6= A ((1− λ)A + λA ⊃ A).
But if A is convex
(1− λ)A + λA = A ∀λ ∈ [0, 1].
![Page 13: Around the Brunn-Minkowski inequality - UniFIweb.math.unifi.it/users/colesant/ricerca/Magdeburgo e Berlino 2015.pdf · Around the Brunn-Minkowski inequality Andrea Colesanti Technische](https://reader031.fdocuments.in/reader031/viewer/2022030820/5b33fad37f8b9a6b548ba38c/html5/thumbnails/13.jpg)
(BM) privileges convex sets
In the inequality
Vn((1− λ)A + λB)1/n ≥ (1− λ)Vn(A)1/n + λVn(B)1/n,
equality holds iff A is convex and B is a homothetic copy of A (upto subsets of volume zero).
Why? If you plug A = B in (BM) in general you don’t get anequality, because
(1− λ)A + λA 6= A
((1− λ)A + λA ⊃ A).
But if A is convex
(1− λ)A + λA = A ∀λ ∈ [0, 1].
![Page 14: Around the Brunn-Minkowski inequality - UniFIweb.math.unifi.it/users/colesant/ricerca/Magdeburgo e Berlino 2015.pdf · Around the Brunn-Minkowski inequality Andrea Colesanti Technische](https://reader031.fdocuments.in/reader031/viewer/2022030820/5b33fad37f8b9a6b548ba38c/html5/thumbnails/14.jpg)
(BM) privileges convex sets
In the inequality
Vn((1− λ)A + λB)1/n ≥ (1− λ)Vn(A)1/n + λVn(B)1/n,
equality holds iff A is convex and B is a homothetic copy of A (upto subsets of volume zero).
Why? If you plug A = B in (BM) in general you don’t get anequality, because
(1− λ)A + λA 6= A ((1− λ)A + λA ⊃ A).
But if A is convex
(1− λ)A + λA = A ∀λ ∈ [0, 1].
![Page 15: Around the Brunn-Minkowski inequality - UniFIweb.math.unifi.it/users/colesant/ricerca/Magdeburgo e Berlino 2015.pdf · Around the Brunn-Minkowski inequality Andrea Colesanti Technische](https://reader031.fdocuments.in/reader031/viewer/2022030820/5b33fad37f8b9a6b548ba38c/html5/thumbnails/15.jpg)
(BM) privileges convex sets
In the inequality
Vn((1− λ)A + λB)1/n ≥ (1− λ)Vn(A)1/n + λVn(B)1/n,
equality holds iff A is convex and B is a homothetic copy of A (upto subsets of volume zero).
Why? If you plug A = B in (BM) in general you don’t get anequality, because
(1− λ)A + λA 6= A ((1− λ)A + λA ⊃ A).
But if A is convex
(1− λ)A + λA = A ∀λ ∈ [0, 1].
![Page 16: Around the Brunn-Minkowski inequality - UniFIweb.math.unifi.it/users/colesant/ricerca/Magdeburgo e Berlino 2015.pdf · Around the Brunn-Minkowski inequality Andrea Colesanti Technische](https://reader031.fdocuments.in/reader031/viewer/2022030820/5b33fad37f8b9a6b548ba38c/html5/thumbnails/16.jpg)
Many equivalent forms
I Classic
Vn((1−λ)A+λB)1/n ≥ (1−λ)Vn(A)1/n +λVn(B)1/n. (BM)
I Elegant
Vn(A + B)1/n ≥ Vn(A)1/n + Vn(B)1/n. (BM∗)
I Multiplicative
Vn((1− λ)A + λB) ≥ Vn(A)1−λVn(B)λ. (BM0)
I Minimal
Vn((1− λ)A + λB) ≥ min{Vn(A),Vn(B)}. (BM−∞)
![Page 17: Around the Brunn-Minkowski inequality - UniFIweb.math.unifi.it/users/colesant/ricerca/Magdeburgo e Berlino 2015.pdf · Around the Brunn-Minkowski inequality Andrea Colesanti Technische](https://reader031.fdocuments.in/reader031/viewer/2022030820/5b33fad37f8b9a6b548ba38c/html5/thumbnails/17.jpg)
Many equivalent forms
I Classic
Vn((1−λ)A+λB)1/n ≥ (1−λ)Vn(A)1/n +λVn(B)1/n. (BM)
I Elegant
Vn(A + B)1/n ≥ Vn(A)1/n + Vn(B)1/n. (BM∗)
I Multiplicative
Vn((1− λ)A + λB) ≥ Vn(A)1−λVn(B)λ. (BM0)
I Minimal
Vn((1− λ)A + λB) ≥ min{Vn(A),Vn(B)}. (BM−∞)
![Page 18: Around the Brunn-Minkowski inequality - UniFIweb.math.unifi.it/users/colesant/ricerca/Magdeburgo e Berlino 2015.pdf · Around the Brunn-Minkowski inequality Andrea Colesanti Technische](https://reader031.fdocuments.in/reader031/viewer/2022030820/5b33fad37f8b9a6b548ba38c/html5/thumbnails/18.jpg)
Many equivalent forms
I Classic
Vn((1−λ)A+λB)1/n ≥ (1−λ)Vn(A)1/n +λVn(B)1/n. (BM)
I Elegant
Vn(A + B)1/n ≥ Vn(A)1/n + Vn(B)1/n. (BM∗)
I Multiplicative
Vn((1− λ)A + λB) ≥ Vn(A)1−λVn(B)λ. (BM0)
I Minimal
Vn((1− λ)A + λB) ≥ min{Vn(A),Vn(B)}. (BM−∞)
![Page 19: Around the Brunn-Minkowski inequality - UniFIweb.math.unifi.it/users/colesant/ricerca/Magdeburgo e Berlino 2015.pdf · Around the Brunn-Minkowski inequality Andrea Colesanti Technische](https://reader031.fdocuments.in/reader031/viewer/2022030820/5b33fad37f8b9a6b548ba38c/html5/thumbnails/19.jpg)
Many equivalent forms
I Classic
Vn((1−λ)A+λB)1/n ≥ (1−λ)Vn(A)1/n +λVn(B)1/n. (BM)
I Elegant
Vn(A + B)1/n ≥ Vn(A)1/n + Vn(B)1/n. (BM∗)
I Multiplicative
Vn((1− λ)A + λB) ≥ Vn(A)1−λVn(B)λ. (BM0)
I Minimal
Vn((1− λ)A + λB) ≥ min{Vn(A),Vn(B)}. (BM−∞)
![Page 20: Around the Brunn-Minkowski inequality - UniFIweb.math.unifi.it/users/colesant/ricerca/Magdeburgo e Berlino 2015.pdf · Around the Brunn-Minkowski inequality Andrea Colesanti Technische](https://reader031.fdocuments.in/reader031/viewer/2022030820/5b33fad37f8b9a6b548ba38c/html5/thumbnails/20.jpg)
Many equivalent forms
I Classic
Vn((1−λ)A+λB)1/n ≥ (1−λ)Vn(A)1/n +λVn(B)1/n. (BM)
I Elegant
Vn(A + B)1/n ≥ Vn(A)1/n + Vn(B)1/n. (BM∗)
I Multiplicative
Vn((1− λ)A + λB) ≥ Vn(A)1−λVn(B)λ. (BM0)
I Minimal
Vn((1− λ)A + λB) ≥ min{Vn(A),Vn(B)}. (BM−∞)
![Page 21: Around the Brunn-Minkowski inequality - UniFIweb.math.unifi.it/users/colesant/ricerca/Magdeburgo e Berlino 2015.pdf · Around the Brunn-Minkowski inequality Andrea Colesanti Technische](https://reader031.fdocuments.in/reader031/viewer/2022030820/5b33fad37f8b9a6b548ba38c/html5/thumbnails/21.jpg)
A general fact about homogeneous functional
Let F be a real-valued functional
I defined on a convex cone C;
I α-homogeneous: F(λx) = λαF(x), ∀x ∈ C, ∀λ > 0 (α > 0);
I non-negative.
F1/α concave ⇔ {F ≥ t} is convex ∀ t.
The last condition (quasi-concavity) is equivalent to
F((1− λ)A + λB) ≥ min{F(A),F(B)} ∀A,B ∈ C, ∀λ ∈ [0, 1].
In our case: C = {compact sets}, F = Vn, α = n.
![Page 22: Around the Brunn-Minkowski inequality - UniFIweb.math.unifi.it/users/colesant/ricerca/Magdeburgo e Berlino 2015.pdf · Around the Brunn-Minkowski inequality Andrea Colesanti Technische](https://reader031.fdocuments.in/reader031/viewer/2022030820/5b33fad37f8b9a6b548ba38c/html5/thumbnails/22.jpg)
A general fact about homogeneous functional
Let F be a real-valued functional
I defined on a convex cone C;
I α-homogeneous: F(λx) = λαF(x), ∀x ∈ C, ∀λ > 0 (α > 0);
I non-negative.
F1/α concave ⇔ {F ≥ t} is convex ∀ t.
The last condition (quasi-concavity) is equivalent to
F((1− λ)A + λB) ≥ min{F(A),F(B)} ∀A,B ∈ C, ∀λ ∈ [0, 1].
In our case: C = {compact sets}, F = Vn, α = n.
![Page 23: Around the Brunn-Minkowski inequality - UniFIweb.math.unifi.it/users/colesant/ricerca/Magdeburgo e Berlino 2015.pdf · Around the Brunn-Minkowski inequality Andrea Colesanti Technische](https://reader031.fdocuments.in/reader031/viewer/2022030820/5b33fad37f8b9a6b548ba38c/html5/thumbnails/23.jpg)
A general fact about homogeneous functional
Let F be a real-valued functional
I defined on a convex cone C;
I α-homogeneous: F(λx) = λαF(x), ∀x ∈ C, ∀λ > 0 (α > 0);
I non-negative.
F1/α concave ⇔ {F ≥ t} is convex ∀ t.
The last condition (quasi-concavity) is equivalent to
F((1− λ)A + λB) ≥ min{F(A),F(B)} ∀A,B ∈ C, ∀λ ∈ [0, 1].
In our case: C = {compact sets}, F = Vn, α = n.
![Page 24: Around the Brunn-Minkowski inequality - UniFIweb.math.unifi.it/users/colesant/ricerca/Magdeburgo e Berlino 2015.pdf · Around the Brunn-Minkowski inequality Andrea Colesanti Technische](https://reader031.fdocuments.in/reader031/viewer/2022030820/5b33fad37f8b9a6b548ba38c/html5/thumbnails/24.jpg)
An “elementary” proof of (BM) - I
Lemma (Prekopa-Leindler inequality). Let
f , g , h : Rn → R+
be measurable functions, and let λ ∈ [0, 1]. Assume that
f ((1− λ)x + λy)) ≥ g(x)(1−λ) h(y)λ ∀ x , y ∈ Rn.
Then ∫Rn
fdz ≥(∫
Rn
gdx
)1−λ (∫Rn
hdy
)λ.
Proof.
I Prove the 1-dimensional case (using just the so-called layercake, or Cavalieri’s, principle);
I the n-dimensional case follows by induction and Fubini’stheorem.
![Page 25: Around the Brunn-Minkowski inequality - UniFIweb.math.unifi.it/users/colesant/ricerca/Magdeburgo e Berlino 2015.pdf · Around the Brunn-Minkowski inequality Andrea Colesanti Technische](https://reader031.fdocuments.in/reader031/viewer/2022030820/5b33fad37f8b9a6b548ba38c/html5/thumbnails/25.jpg)
An “elementary” proof of (BM) - I
Lemma (Prekopa-Leindler inequality).
Let
f , g , h : Rn → R+
be measurable functions, and let λ ∈ [0, 1]. Assume that
f ((1− λ)x + λy)) ≥ g(x)(1−λ) h(y)λ ∀ x , y ∈ Rn.
Then ∫Rn
fdz ≥(∫
Rn
gdx
)1−λ (∫Rn
hdy
)λ.
Proof.
I Prove the 1-dimensional case (using just the so-called layercake, or Cavalieri’s, principle);
I the n-dimensional case follows by induction and Fubini’stheorem.
![Page 26: Around the Brunn-Minkowski inequality - UniFIweb.math.unifi.it/users/colesant/ricerca/Magdeburgo e Berlino 2015.pdf · Around the Brunn-Minkowski inequality Andrea Colesanti Technische](https://reader031.fdocuments.in/reader031/viewer/2022030820/5b33fad37f8b9a6b548ba38c/html5/thumbnails/26.jpg)
An “elementary” proof of (BM) - I
Lemma (Prekopa-Leindler inequality). Let
f , g , h : Rn → R+
be measurable functions,
and let λ ∈ [0, 1]. Assume that
f ((1− λ)x + λy)) ≥ g(x)(1−λ) h(y)λ ∀ x , y ∈ Rn.
Then ∫Rn
fdz ≥(∫
Rn
gdx
)1−λ (∫Rn
hdy
)λ.
Proof.
I Prove the 1-dimensional case (using just the so-called layercake, or Cavalieri’s, principle);
I the n-dimensional case follows by induction and Fubini’stheorem.
![Page 27: Around the Brunn-Minkowski inequality - UniFIweb.math.unifi.it/users/colesant/ricerca/Magdeburgo e Berlino 2015.pdf · Around the Brunn-Minkowski inequality Andrea Colesanti Technische](https://reader031.fdocuments.in/reader031/viewer/2022030820/5b33fad37f8b9a6b548ba38c/html5/thumbnails/27.jpg)
An “elementary” proof of (BM) - I
Lemma (Prekopa-Leindler inequality). Let
f , g , h : Rn → R+
be measurable functions, and let λ ∈ [0, 1].
Assume that
f ((1− λ)x + λy)) ≥ g(x)(1−λ) h(y)λ ∀ x , y ∈ Rn.
Then ∫Rn
fdz ≥(∫
Rn
gdx
)1−λ (∫Rn
hdy
)λ.
Proof.
I Prove the 1-dimensional case (using just the so-called layercake, or Cavalieri’s, principle);
I the n-dimensional case follows by induction and Fubini’stheorem.
![Page 28: Around the Brunn-Minkowski inequality - UniFIweb.math.unifi.it/users/colesant/ricerca/Magdeburgo e Berlino 2015.pdf · Around the Brunn-Minkowski inequality Andrea Colesanti Technische](https://reader031.fdocuments.in/reader031/viewer/2022030820/5b33fad37f8b9a6b548ba38c/html5/thumbnails/28.jpg)
An “elementary” proof of (BM) - I
Lemma (Prekopa-Leindler inequality). Let
f , g , h : Rn → R+
be measurable functions, and let λ ∈ [0, 1]. Assume that
f ((1− λ)x + λy)) ≥ g(x)(1−λ) h(y)λ ∀ x , y ∈ Rn.
Then ∫Rn
fdz ≥(∫
Rn
gdx
)1−λ (∫Rn
hdy
)λ.
Proof.
I Prove the 1-dimensional case (using just the so-called layercake, or Cavalieri’s, principle);
I the n-dimensional case follows by induction and Fubini’stheorem.
![Page 29: Around the Brunn-Minkowski inequality - UniFIweb.math.unifi.it/users/colesant/ricerca/Magdeburgo e Berlino 2015.pdf · Around the Brunn-Minkowski inequality Andrea Colesanti Technische](https://reader031.fdocuments.in/reader031/viewer/2022030820/5b33fad37f8b9a6b548ba38c/html5/thumbnails/29.jpg)
An “elementary” proof of (BM) - I
Lemma (Prekopa-Leindler inequality). Let
f , g , h : Rn → R+
be measurable functions, and let λ ∈ [0, 1]. Assume that
f ((1− λ)x + λy)) ≥ g(x)(1−λ) h(y)λ ∀ x , y ∈ Rn.
Then ∫Rn
fdz ≥(∫
Rn
gdx
)1−λ (∫Rn
hdy
)λ.
Proof.
I Prove the 1-dimensional case (using just the so-called layercake, or Cavalieri’s, principle);
I the n-dimensional case follows by induction and Fubini’stheorem.
![Page 30: Around the Brunn-Minkowski inequality - UniFIweb.math.unifi.it/users/colesant/ricerca/Magdeburgo e Berlino 2015.pdf · Around the Brunn-Minkowski inequality Andrea Colesanti Technische](https://reader031.fdocuments.in/reader031/viewer/2022030820/5b33fad37f8b9a6b548ba38c/html5/thumbnails/30.jpg)
An “elementary” proof of (BM) - I
Lemma (Prekopa-Leindler inequality). Let
f , g , h : Rn → R+
be measurable functions, and let λ ∈ [0, 1]. Assume that
f ((1− λ)x + λy)) ≥ g(x)(1−λ) h(y)λ ∀ x , y ∈ Rn.
Then ∫Rn
fdz ≥(∫
Rn
gdx
)1−λ (∫Rn
hdy
)λ.
Proof.
I Prove the 1-dimensional case (using just the so-called layercake, or Cavalieri’s, principle);
I the n-dimensional case follows by induction and Fubini’stheorem.
![Page 31: Around the Brunn-Minkowski inequality - UniFIweb.math.unifi.it/users/colesant/ricerca/Magdeburgo e Berlino 2015.pdf · Around the Brunn-Minkowski inequality Andrea Colesanti Technische](https://reader031.fdocuments.in/reader031/viewer/2022030820/5b33fad37f8b9a6b548ba38c/html5/thumbnails/31.jpg)
An “elementary” proof of (BM) - I
Lemma (Prekopa-Leindler inequality). Let
f , g , h : Rn → R+
be measurable functions, and let λ ∈ [0, 1]. Assume that
f ((1− λ)x + λy)) ≥ g(x)(1−λ) h(y)λ ∀ x , y ∈ Rn.
Then ∫Rn
fdz ≥(∫
Rn
gdx
)1−λ (∫Rn
hdy
)λ.
Proof.
I Prove the 1-dimensional case (using just the so-called layercake, or Cavalieri’s, principle);
I the n-dimensional case follows by induction and Fubini’stheorem.
![Page 32: Around the Brunn-Minkowski inequality - UniFIweb.math.unifi.it/users/colesant/ricerca/Magdeburgo e Berlino 2015.pdf · Around the Brunn-Minkowski inequality Andrea Colesanti Technische](https://reader031.fdocuments.in/reader031/viewer/2022030820/5b33fad37f8b9a6b548ba38c/html5/thumbnails/32.jpg)
A proof of (BM) - II
Given A,B ⊂ Rn and λ ∈ [0, 1], let
f = characteristic function of (1− λ)A + λB,
g = charact. function of A, h = charact. function of B.
Then:
f ((1− λ)x + λy)) ≥ g(x)(1−λ) h(y)λ ∀ x , y ∈ Rn.
By Prekopa-Leindler inequality
Vn((1− λ)A + λB) =
∫Rn
fdz
≥(∫
Rn
gdx
)1−λ (∫Rn
hdy
)λ= Vn(A)1−λVn(B)λ,
i.e. the multiplicative form of (BM).
![Page 33: Around the Brunn-Minkowski inequality - UniFIweb.math.unifi.it/users/colesant/ricerca/Magdeburgo e Berlino 2015.pdf · Around the Brunn-Minkowski inequality Andrea Colesanti Technische](https://reader031.fdocuments.in/reader031/viewer/2022030820/5b33fad37f8b9a6b548ba38c/html5/thumbnails/33.jpg)
A proof of (BM) - IIGiven A,B ⊂ Rn and λ ∈ [0, 1], let
f = characteristic function of (1− λ)A + λB,
g = charact. function of A, h = charact. function of B.
Then:
f ((1− λ)x + λy)) ≥ g(x)(1−λ) h(y)λ ∀ x , y ∈ Rn.
By Prekopa-Leindler inequality
Vn((1− λ)A + λB) =
∫Rn
fdz
≥(∫
Rn
gdx
)1−λ (∫Rn
hdy
)λ= Vn(A)1−λVn(B)λ,
i.e. the multiplicative form of (BM).
![Page 34: Around the Brunn-Minkowski inequality - UniFIweb.math.unifi.it/users/colesant/ricerca/Magdeburgo e Berlino 2015.pdf · Around the Brunn-Minkowski inequality Andrea Colesanti Technische](https://reader031.fdocuments.in/reader031/viewer/2022030820/5b33fad37f8b9a6b548ba38c/html5/thumbnails/34.jpg)
A proof of (BM) - IIGiven A,B ⊂ Rn and λ ∈ [0, 1], let
f = characteristic function of (1− λ)A + λB,
g = charact. function of A, h = charact. function of B.
Then:
f ((1− λ)x + λy)) ≥ g(x)(1−λ) h(y)λ ∀ x , y ∈ Rn.
By Prekopa-Leindler inequality
Vn((1− λ)A + λB) =
∫Rn
fdz
≥(∫
Rn
gdx
)1−λ (∫Rn
hdy
)λ= Vn(A)1−λVn(B)λ,
i.e. the multiplicative form of (BM).
![Page 35: Around the Brunn-Minkowski inequality - UniFIweb.math.unifi.it/users/colesant/ricerca/Magdeburgo e Berlino 2015.pdf · Around the Brunn-Minkowski inequality Andrea Colesanti Technische](https://reader031.fdocuments.in/reader031/viewer/2022030820/5b33fad37f8b9a6b548ba38c/html5/thumbnails/35.jpg)
A proof of (BM) - IIGiven A,B ⊂ Rn and λ ∈ [0, 1], let
f = characteristic function of (1− λ)A + λB,
g = charact. function of A, h = charact. function of B.
Then:
f ((1− λ)x + λy)) ≥ g(x)(1−λ) h(y)λ ∀ x , y ∈ Rn.
By Prekopa-Leindler inequality
Vn((1− λ)A + λB) =
∫Rn
fdz
≥(∫
Rn
gdx
)1−λ (∫Rn
hdy
)λ= Vn(A)1−λVn(B)λ,
i.e. the multiplicative form of (BM).
![Page 36: Around the Brunn-Minkowski inequality - UniFIweb.math.unifi.it/users/colesant/ricerca/Magdeburgo e Berlino 2015.pdf · Around the Brunn-Minkowski inequality Andrea Colesanti Technische](https://reader031.fdocuments.in/reader031/viewer/2022030820/5b33fad37f8b9a6b548ba38c/html5/thumbnails/36.jpg)
A proof of (BM) - IIGiven A,B ⊂ Rn and λ ∈ [0, 1], let
f = characteristic function of (1− λ)A + λB,
g = charact. function of A, h = charact. function of B.
Then:
f ((1− λ)x + λy)) ≥ g(x)(1−λ) h(y)λ ∀ x , y ∈ Rn.
By Prekopa-Leindler inequality
Vn((1− λ)A + λB) =
∫Rn
fdz
≥(∫
Rn
gdx
)1−λ (∫Rn
hdy
)λ= Vn(A)1−λVn(B)λ,
i.e. the multiplicative form of (BM).
![Page 37: Around the Brunn-Minkowski inequality - UniFIweb.math.unifi.it/users/colesant/ricerca/Magdeburgo e Berlino 2015.pdf · Around the Brunn-Minkowski inequality Andrea Colesanti Technische](https://reader031.fdocuments.in/reader031/viewer/2022030820/5b33fad37f8b9a6b548ba38c/html5/thumbnails/37.jpg)
A proof of (BM) - IIGiven A,B ⊂ Rn and λ ∈ [0, 1], let
f = characteristic function of (1− λ)A + λB,
g = charact. function of A, h = charact. function of B.
Then:
f ((1− λ)x + λy)) ≥ g(x)(1−λ) h(y)λ ∀ x , y ∈ Rn.
By Prekopa-Leindler inequality
Vn((1− λ)A + λB) =
∫Rn
fdz
≥(∫
Rn
gdx
)1−λ (∫Rn
hdy
)λ= Vn(A)1−λVn(B)λ,
i.e. the multiplicative form of (BM).
![Page 38: Around the Brunn-Minkowski inequality - UniFIweb.math.unifi.it/users/colesant/ricerca/Magdeburgo e Berlino 2015.pdf · Around the Brunn-Minkowski inequality Andrea Colesanti Technische](https://reader031.fdocuments.in/reader031/viewer/2022030820/5b33fad37f8b9a6b548ba38c/html5/thumbnails/38.jpg)
A proof of (BM) - IIGiven A,B ⊂ Rn and λ ∈ [0, 1], let
f = characteristic function of (1− λ)A + λB,
g = charact. function of A, h = charact. function of B.
Then:
f ((1− λ)x + λy)) ≥ g(x)(1−λ) h(y)λ ∀ x , y ∈ Rn.
By Prekopa-Leindler inequality
Vn((1− λ)A + λB) =
∫Rn
fdz
≥(∫
Rn
gdx
)1−λ (∫Rn
hdy
)λ
= Vn(A)1−λVn(B)λ,
i.e. the multiplicative form of (BM).
![Page 39: Around the Brunn-Minkowski inequality - UniFIweb.math.unifi.it/users/colesant/ricerca/Magdeburgo e Berlino 2015.pdf · Around the Brunn-Minkowski inequality Andrea Colesanti Technische](https://reader031.fdocuments.in/reader031/viewer/2022030820/5b33fad37f8b9a6b548ba38c/html5/thumbnails/39.jpg)
A proof of (BM) - IIGiven A,B ⊂ Rn and λ ∈ [0, 1], let
f = characteristic function of (1− λ)A + λB,
g = charact. function of A, h = charact. function of B.
Then:
f ((1− λ)x + λy)) ≥ g(x)(1−λ) h(y)λ ∀ x , y ∈ Rn.
By Prekopa-Leindler inequality
Vn((1− λ)A + λB) =
∫Rn
fdz
≥(∫
Rn
gdx
)1−λ (∫Rn
hdy
)λ= Vn(A)1−λVn(B)λ,
i.e. the multiplicative form of (BM).
![Page 40: Around the Brunn-Minkowski inequality - UniFIweb.math.unifi.it/users/colesant/ricerca/Magdeburgo e Berlino 2015.pdf · Around the Brunn-Minkowski inequality Andrea Colesanti Technische](https://reader031.fdocuments.in/reader031/viewer/2022030820/5b33fad37f8b9a6b548ba38c/html5/thumbnails/40.jpg)
A proof of (BM) - IIGiven A,B ⊂ Rn and λ ∈ [0, 1], let
f = characteristic function of (1− λ)A + λB,
g = charact. function of A, h = charact. function of B.
Then:
f ((1− λ)x + λy)) ≥ g(x)(1−λ) h(y)λ ∀ x , y ∈ Rn.
By Prekopa-Leindler inequality
Vn((1− λ)A + λB) =
∫Rn
fdz
≥(∫
Rn
gdx
)1−λ (∫Rn
hdy
)λ= Vn(A)1−λVn(B)λ,
i.e. the multiplicative form of (BM).
![Page 41: Around the Brunn-Minkowski inequality - UniFIweb.math.unifi.it/users/colesant/ricerca/Magdeburgo e Berlino 2015.pdf · Around the Brunn-Minkowski inequality Andrea Colesanti Technische](https://reader031.fdocuments.in/reader031/viewer/2022030820/5b33fad37f8b9a6b548ba38c/html5/thumbnails/41.jpg)
The isoperimetric inequality
Thm. Among all subsets of Rn with given perimeter, the ballhaving such perimeter maximizes the volume.
Equivalently,
Vn(A)n−1n ≤ c(n)Hn−1(∂A)
for every set A (with sufficiently smooth boundary), where c(n) isa constant and Hn−1 is the (n − 1)-dimensional Hausdorffmeasure. Equality is attained when A is a ball.
![Page 42: Around the Brunn-Minkowski inequality - UniFIweb.math.unifi.it/users/colesant/ricerca/Magdeburgo e Berlino 2015.pdf · Around the Brunn-Minkowski inequality Andrea Colesanti Technische](https://reader031.fdocuments.in/reader031/viewer/2022030820/5b33fad37f8b9a6b548ba38c/html5/thumbnails/42.jpg)
The isoperimetric inequality
Thm. Among all subsets of Rn with given perimeter, the ballhaving such perimeter maximizes the volume.
Equivalently,
Vn(A)n−1n ≤ c(n)Hn−1(∂A)
for every set A (with sufficiently smooth boundary), where c(n) isa constant and Hn−1 is the (n − 1)-dimensional Hausdorffmeasure. Equality is attained when A is a ball.
![Page 43: Around the Brunn-Minkowski inequality - UniFIweb.math.unifi.it/users/colesant/ricerca/Magdeburgo e Berlino 2015.pdf · Around the Brunn-Minkowski inequality Andrea Colesanti Technische](https://reader031.fdocuments.in/reader031/viewer/2022030820/5b33fad37f8b9a6b548ba38c/html5/thumbnails/43.jpg)
The isoperimetric inequality
Thm. Among all subsets of Rn with given perimeter, the ballhaving such perimeter maximizes the volume.
Equivalently,
Vn(A)n−1n ≤ c(n)Hn−1(∂A)
for every set A (with sufficiently smooth boundary), where c(n) isa constant and Hn−1 is the (n − 1)-dimensional Hausdorffmeasure. Equality is attained when A is a ball.
![Page 44: Around the Brunn-Minkowski inequality - UniFIweb.math.unifi.it/users/colesant/ricerca/Magdeburgo e Berlino 2015.pdf · Around the Brunn-Minkowski inequality Andrea Colesanti Technische](https://reader031.fdocuments.in/reader031/viewer/2022030820/5b33fad37f8b9a6b548ba38c/html5/thumbnails/44.jpg)
(BM) ⇒ isoperimetric inequality
Let A ⊂ Rn be a bounded domain with C 1 boundary. Then
Hn−1(∂A) = limε→0+
Vn(Aε)− Vn(A)
ε,
where
Aε = {x ∈ Rn : dist(x ,A) ≤ ε}
= A + εB,
andB = {x ∈ Rn : ‖x‖ ≤ 1} = unit ball.
Hence
Hn−1(∂A) = limε→0+
Vn(A + εB)− Vn(A)
ε.
![Page 45: Around the Brunn-Minkowski inequality - UniFIweb.math.unifi.it/users/colesant/ricerca/Magdeburgo e Berlino 2015.pdf · Around the Brunn-Minkowski inequality Andrea Colesanti Technische](https://reader031.fdocuments.in/reader031/viewer/2022030820/5b33fad37f8b9a6b548ba38c/html5/thumbnails/45.jpg)
(BM) ⇒ isoperimetric inequality
Let A ⊂ Rn be a bounded domain with C 1 boundary. Then
Hn−1(∂A) = limε→0+
Vn(Aε)− Vn(A)
ε,
where
Aε = {x ∈ Rn : dist(x ,A) ≤ ε}= A + εB,
andB = {x ∈ Rn : ‖x‖ ≤ 1} = unit ball.
Hence
Hn−1(∂A) = limε→0+
Vn(A + εB)− Vn(A)
ε.
![Page 46: Around the Brunn-Minkowski inequality - UniFIweb.math.unifi.it/users/colesant/ricerca/Magdeburgo e Berlino 2015.pdf · Around the Brunn-Minkowski inequality Andrea Colesanti Technische](https://reader031.fdocuments.in/reader031/viewer/2022030820/5b33fad37f8b9a6b548ba38c/html5/thumbnails/46.jpg)
(BM) ⇒ isoperimetric inequality
Let A ⊂ Rn be a bounded domain with C 1 boundary. Then
Hn−1(∂A) = limε→0+
Vn(Aε)− Vn(A)
ε,
where
Aε = {x ∈ Rn : dist(x ,A) ≤ ε}= A + εB,
andB = {x ∈ Rn : ‖x‖ ≤ 1} = unit ball.
Hence
Hn−1(∂A) = limε→0+
Vn(A + εB)− Vn(A)
ε.
![Page 47: Around the Brunn-Minkowski inequality - UniFIweb.math.unifi.it/users/colesant/ricerca/Magdeburgo e Berlino 2015.pdf · Around the Brunn-Minkowski inequality Andrea Colesanti Technische](https://reader031.fdocuments.in/reader031/viewer/2022030820/5b33fad37f8b9a6b548ba38c/html5/thumbnails/47.jpg)
Proof of the isoperimetric inequality
Hn−1(∂A) = limε→0+
Vn(A + εB)− Vn(A)
ε.
By (BM), for every ε > 0
Vn(A + εB)1/n ≥ Vn(A)1/n + Vn(εB)1/n = Vn(A)1/n + εVn(B)1/n.
Vn(B)1/n ≤ limε→0+
Vn(A + εB)1/n − Vn(A)1/n
ε
=1
nVn(A)
1−nn Hn−1(∂A).
Vn(A)n−1n ≤ 1
nVn(B)1/nHn−1(∂A) = c(n)Hn−1(∂A).
When A is a ball this becomes an equality.
![Page 48: Around the Brunn-Minkowski inequality - UniFIweb.math.unifi.it/users/colesant/ricerca/Magdeburgo e Berlino 2015.pdf · Around the Brunn-Minkowski inequality Andrea Colesanti Technische](https://reader031.fdocuments.in/reader031/viewer/2022030820/5b33fad37f8b9a6b548ba38c/html5/thumbnails/48.jpg)
Proof of the isoperimetric inequality
Hn−1(∂A) = limε→0+
Vn(A + εB)− Vn(A)
ε.
By (BM), for every ε > 0
Vn(A + εB)1/n ≥ Vn(A)1/n + Vn(εB)1/n = Vn(A)1/n + εVn(B)1/n.
Vn(B)1/n ≤ limε→0+
Vn(A + εB)1/n − Vn(A)1/n
ε
=1
nVn(A)
1−nn Hn−1(∂A).
Vn(A)n−1n ≤ 1
nVn(B)1/nHn−1(∂A) = c(n)Hn−1(∂A).
When A is a ball this becomes an equality.
![Page 49: Around the Brunn-Minkowski inequality - UniFIweb.math.unifi.it/users/colesant/ricerca/Magdeburgo e Berlino 2015.pdf · Around the Brunn-Minkowski inequality Andrea Colesanti Technische](https://reader031.fdocuments.in/reader031/viewer/2022030820/5b33fad37f8b9a6b548ba38c/html5/thumbnails/49.jpg)
Proof of the isoperimetric inequality
Hn−1(∂A) = limε→0+
Vn(A + εB)− Vn(A)
ε.
By (BM), for every ε > 0
Vn(A + εB)1/n ≥ Vn(A)1/n + Vn(εB)1/n
= Vn(A)1/n + εVn(B)1/n.
Vn(B)1/n ≤ limε→0+
Vn(A + εB)1/n − Vn(A)1/n
ε
=1
nVn(A)
1−nn Hn−1(∂A).
Vn(A)n−1n ≤ 1
nVn(B)1/nHn−1(∂A) = c(n)Hn−1(∂A).
When A is a ball this becomes an equality.
![Page 50: Around the Brunn-Minkowski inequality - UniFIweb.math.unifi.it/users/colesant/ricerca/Magdeburgo e Berlino 2015.pdf · Around the Brunn-Minkowski inequality Andrea Colesanti Technische](https://reader031.fdocuments.in/reader031/viewer/2022030820/5b33fad37f8b9a6b548ba38c/html5/thumbnails/50.jpg)
Proof of the isoperimetric inequality
Hn−1(∂A) = limε→0+
Vn(A + εB)− Vn(A)
ε.
By (BM), for every ε > 0
Vn(A + εB)1/n ≥ Vn(A)1/n + Vn(εB)1/n = Vn(A)1/n + εVn(B)1/n.
Vn(B)1/n ≤ limε→0+
Vn(A + εB)1/n − Vn(A)1/n
ε
=1
nVn(A)
1−nn Hn−1(∂A).
Vn(A)n−1n ≤ 1
nVn(B)1/nHn−1(∂A) = c(n)Hn−1(∂A).
When A is a ball this becomes an equality.
![Page 51: Around the Brunn-Minkowski inequality - UniFIweb.math.unifi.it/users/colesant/ricerca/Magdeburgo e Berlino 2015.pdf · Around the Brunn-Minkowski inequality Andrea Colesanti Technische](https://reader031.fdocuments.in/reader031/viewer/2022030820/5b33fad37f8b9a6b548ba38c/html5/thumbnails/51.jpg)
Proof of the isoperimetric inequality
Hn−1(∂A) = limε→0+
Vn(A + εB)− Vn(A)
ε.
By (BM), for every ε > 0
Vn(A + εB)1/n ≥ Vn(A)1/n + Vn(εB)1/n = Vn(A)1/n + εVn(B)1/n.
Vn(B)1/n ≤ limε→0+
Vn(A + εB)1/n − Vn(A)1/n
ε
=1
nVn(A)
1−nn Hn−1(∂A).
Vn(A)n−1n ≤ 1
nVn(B)1/nHn−1(∂A) = c(n)Hn−1(∂A).
When A is a ball this becomes an equality.
![Page 52: Around the Brunn-Minkowski inequality - UniFIweb.math.unifi.it/users/colesant/ricerca/Magdeburgo e Berlino 2015.pdf · Around the Brunn-Minkowski inequality Andrea Colesanti Technische](https://reader031.fdocuments.in/reader031/viewer/2022030820/5b33fad37f8b9a6b548ba38c/html5/thumbnails/52.jpg)
Proof of the isoperimetric inequality
Hn−1(∂A) = limε→0+
Vn(A + εB)− Vn(A)
ε.
By (BM), for every ε > 0
Vn(A + εB)1/n ≥ Vn(A)1/n + Vn(εB)1/n = Vn(A)1/n + εVn(B)1/n.
Vn(B)1/n ≤ limε→0+
Vn(A + εB)1/n − Vn(A)1/n
ε
=1
nVn(A)
1−nn Hn−1(∂A).
Vn(A)n−1n ≤ 1
nVn(B)1/nHn−1(∂A) = c(n)Hn−1(∂A).
When A is a ball this becomes an equality.
![Page 53: Around the Brunn-Minkowski inequality - UniFIweb.math.unifi.it/users/colesant/ricerca/Magdeburgo e Berlino 2015.pdf · Around the Brunn-Minkowski inequality Andrea Colesanti Technische](https://reader031.fdocuments.in/reader031/viewer/2022030820/5b33fad37f8b9a6b548ba38c/html5/thumbnails/53.jpg)
Proof of the isoperimetric inequality
Hn−1(∂A) = limε→0+
Vn(A + εB)− Vn(A)
ε.
By (BM), for every ε > 0
Vn(A + εB)1/n ≥ Vn(A)1/n + Vn(εB)1/n = Vn(A)1/n + εVn(B)1/n.
Vn(B)1/n ≤ limε→0+
Vn(A + εB)1/n − Vn(A)1/n
ε
=1
nVn(A)
1−nn Hn−1(∂A).
Vn(A)n−1n ≤ 1
nVn(B)1/nHn−1(∂A)
= c(n)Hn−1(∂A).
When A is a ball this becomes an equality.
![Page 54: Around the Brunn-Minkowski inequality - UniFIweb.math.unifi.it/users/colesant/ricerca/Magdeburgo e Berlino 2015.pdf · Around the Brunn-Minkowski inequality Andrea Colesanti Technische](https://reader031.fdocuments.in/reader031/viewer/2022030820/5b33fad37f8b9a6b548ba38c/html5/thumbnails/54.jpg)
Proof of the isoperimetric inequality
Hn−1(∂A) = limε→0+
Vn(A + εB)− Vn(A)
ε.
By (BM), for every ε > 0
Vn(A + εB)1/n ≥ Vn(A)1/n + Vn(εB)1/n = Vn(A)1/n + εVn(B)1/n.
Vn(B)1/n ≤ limε→0+
Vn(A + εB)1/n − Vn(A)1/n
ε
=1
nVn(A)
1−nn Hn−1(∂A).
Vn(A)n−1n ≤ 1
nVn(B)1/nHn−1(∂A) = c(n)Hn−1(∂A).
When A is a ball this becomes an equality.
![Page 55: Around the Brunn-Minkowski inequality - UniFIweb.math.unifi.it/users/colesant/ricerca/Magdeburgo e Berlino 2015.pdf · Around the Brunn-Minkowski inequality Andrea Colesanti Technische](https://reader031.fdocuments.in/reader031/viewer/2022030820/5b33fad37f8b9a6b548ba38c/html5/thumbnails/55.jpg)
Proof of the isoperimetric inequality
Hn−1(∂A) = limε→0+
Vn(A + εB)− Vn(A)
ε.
By (BM), for every ε > 0
Vn(A + εB)1/n ≥ Vn(A)1/n + Vn(εB)1/n = Vn(A)1/n + εVn(B)1/n.
Vn(B)1/n ≤ limε→0+
Vn(A + εB)1/n − Vn(A)1/n
ε
=1
nVn(A)
1−nn Hn−1(∂A).
Vn(A)n−1n ≤ 1
nVn(B)1/nHn−1(∂A) = c(n)Hn−1(∂A).
When A is a ball this becomes an equality.
![Page 56: Around the Brunn-Minkowski inequality - UniFIweb.math.unifi.it/users/colesant/ricerca/Magdeburgo e Berlino 2015.pdf · Around the Brunn-Minkowski inequality Andrea Colesanti Technische](https://reader031.fdocuments.in/reader031/viewer/2022030820/5b33fad37f8b9a6b548ba38c/html5/thumbnails/56.jpg)
The infinitesimal form of (BM)
I By the Brunn-Minkowski inequality V1/nn is a concave
functional.
I Hence the second variation (or second differential) of V1/nn
(whatever that means) must be negative semidefinite:
D2(V1/nn ) ≤ 0.
I If we restrict our attention to convex sets, this fact amountsto a class of functional inequalities of Poincare type on theunit sphere; a prototype is∫
Sn−1
φ2dHn−1 ≤ c(n)
∫Sn−1
|∇φ|2dHn−1,
∀ φ ∈ C 1(Sn−1), verifying some zero-mean condition.
![Page 57: Around the Brunn-Minkowski inequality - UniFIweb.math.unifi.it/users/colesant/ricerca/Magdeburgo e Berlino 2015.pdf · Around the Brunn-Minkowski inequality Andrea Colesanti Technische](https://reader031.fdocuments.in/reader031/viewer/2022030820/5b33fad37f8b9a6b548ba38c/html5/thumbnails/57.jpg)
The infinitesimal form of (BM)
I By the Brunn-Minkowski inequality V1/nn is a concave
functional.
I Hence the second variation (or second differential) of V1/nn
(whatever that means) must be negative semidefinite:
D2(V1/nn ) ≤ 0.
I If we restrict our attention to convex sets, this fact amountsto a class of functional inequalities of Poincare type on theunit sphere; a prototype is∫
Sn−1
φ2dHn−1 ≤ c(n)
∫Sn−1
|∇φ|2dHn−1,
∀ φ ∈ C 1(Sn−1), verifying some zero-mean condition.
![Page 58: Around the Brunn-Minkowski inequality - UniFIweb.math.unifi.it/users/colesant/ricerca/Magdeburgo e Berlino 2015.pdf · Around the Brunn-Minkowski inequality Andrea Colesanti Technische](https://reader031.fdocuments.in/reader031/viewer/2022030820/5b33fad37f8b9a6b548ba38c/html5/thumbnails/58.jpg)
The infinitesimal form of (BM)
I By the Brunn-Minkowski inequality V1/nn is a concave
functional.
I Hence the second variation (or second differential) of V1/nn
(whatever that means) must be negative semidefinite:
D2(V1/nn ) ≤ 0.
I If we restrict our attention to convex sets, this fact amountsto a class of functional inequalities of Poincare type on theunit sphere; a prototype is∫
Sn−1
φ2dHn−1 ≤ c(n)
∫Sn−1
|∇φ|2dHn−1,
∀ φ ∈ C 1(Sn−1), verifying some zero-mean condition.
![Page 59: Around the Brunn-Minkowski inequality - UniFIweb.math.unifi.it/users/colesant/ricerca/Magdeburgo e Berlino 2015.pdf · Around the Brunn-Minkowski inequality Andrea Colesanti Technische](https://reader031.fdocuments.in/reader031/viewer/2022030820/5b33fad37f8b9a6b548ba38c/html5/thumbnails/59.jpg)
The infinitesimal form of (BM)
I By the Brunn-Minkowski inequality V1/nn is a concave
functional.
I Hence the second variation (or second differential) of V1/nn
(whatever that means) must be negative semidefinite:
D2(V1/nn ) ≤ 0.
I If we restrict our attention to convex sets, this fact amountsto a class of functional inequalities of Poincare type on theunit sphere;
a prototype is∫Sn−1
φ2dHn−1 ≤ c(n)
∫Sn−1
|∇φ|2dHn−1,
∀ φ ∈ C 1(Sn−1), verifying some zero-mean condition.
![Page 60: Around the Brunn-Minkowski inequality - UniFIweb.math.unifi.it/users/colesant/ricerca/Magdeburgo e Berlino 2015.pdf · Around the Brunn-Minkowski inequality Andrea Colesanti Technische](https://reader031.fdocuments.in/reader031/viewer/2022030820/5b33fad37f8b9a6b548ba38c/html5/thumbnails/60.jpg)
The infinitesimal form of (BM)
I By the Brunn-Minkowski inequality V1/nn is a concave
functional.
I Hence the second variation (or second differential) of V1/nn
(whatever that means) must be negative semidefinite:
D2(V1/nn ) ≤ 0.
I If we restrict our attention to convex sets, this fact amountsto a class of functional inequalities of Poincare type on theunit sphere; a prototype is∫
Sn−1
φ2dHn−1 ≤ c(n)
∫Sn−1
|∇φ|2dHn−1,
∀ φ ∈ C 1(Sn−1), verifying some zero-mean condition.
![Page 61: Around the Brunn-Minkowski inequality - UniFIweb.math.unifi.it/users/colesant/ricerca/Magdeburgo e Berlino 2015.pdf · Around the Brunn-Minkowski inequality Andrea Colesanti Technische](https://reader031.fdocuments.in/reader031/viewer/2022030820/5b33fad37f8b9a6b548ba38c/html5/thumbnails/61.jpg)
The infinitesimal form of (BM)
I By the Brunn-Minkowski inequality V1/nn is a concave
functional.
I Hence the second variation (or second differential) of V1/nn
(whatever that means) must be negative semidefinite:
D2(V1/nn ) ≤ 0.
I If we restrict our attention to convex sets, this fact amountsto a class of functional inequalities of Poincare type on theunit sphere; a prototype is∫
Sn−1
φ2dHn−1 ≤ c(n)
∫Sn−1
|∇φ|2dHn−1,
∀ φ ∈ C 1(Sn−1), verifying some zero-mean condition.
![Page 62: Around the Brunn-Minkowski inequality - UniFIweb.math.unifi.it/users/colesant/ricerca/Magdeburgo e Berlino 2015.pdf · Around the Brunn-Minkowski inequality Andrea Colesanti Technische](https://reader031.fdocuments.in/reader031/viewer/2022030820/5b33fad37f8b9a6b548ba38c/html5/thumbnails/62.jpg)
Convex bodies
From now on we will only consider a special type of compact sets:convex bodies.A convex body is a compact convex subset of Rn. We set
Kn = {convex bodies in Rn}.
Kn is closed under addition and dilations: given K , L ∈ Kn andα, β ≥ 0,
αK + βL ∈ Kn.
The Brunn-Minkowski inequality holds in particular in Kn.
![Page 63: Around the Brunn-Minkowski inequality - UniFIweb.math.unifi.it/users/colesant/ricerca/Magdeburgo e Berlino 2015.pdf · Around the Brunn-Minkowski inequality Andrea Colesanti Technische](https://reader031.fdocuments.in/reader031/viewer/2022030820/5b33fad37f8b9a6b548ba38c/html5/thumbnails/63.jpg)
Convex bodies
From now on we will only consider a special type of compact sets:convex bodies.
A convex body is a compact convex subset of Rn. We set
Kn = {convex bodies in Rn}.
Kn is closed under addition and dilations: given K , L ∈ Kn andα, β ≥ 0,
αK + βL ∈ Kn.
The Brunn-Minkowski inequality holds in particular in Kn.
![Page 64: Around the Brunn-Minkowski inequality - UniFIweb.math.unifi.it/users/colesant/ricerca/Magdeburgo e Berlino 2015.pdf · Around the Brunn-Minkowski inequality Andrea Colesanti Technische](https://reader031.fdocuments.in/reader031/viewer/2022030820/5b33fad37f8b9a6b548ba38c/html5/thumbnails/64.jpg)
Convex bodies
From now on we will only consider a special type of compact sets:convex bodies.A convex body is a compact convex subset of Rn. We set
Kn = {convex bodies in Rn}.
Kn is closed under addition and dilations: given K , L ∈ Kn andα, β ≥ 0,
αK + βL ∈ Kn.
The Brunn-Minkowski inequality holds in particular in Kn.
![Page 65: Around the Brunn-Minkowski inequality - UniFIweb.math.unifi.it/users/colesant/ricerca/Magdeburgo e Berlino 2015.pdf · Around the Brunn-Minkowski inequality Andrea Colesanti Technische](https://reader031.fdocuments.in/reader031/viewer/2022030820/5b33fad37f8b9a6b548ba38c/html5/thumbnails/65.jpg)
Convex bodies
From now on we will only consider a special type of compact sets:convex bodies.A convex body is a compact convex subset of Rn. We set
Kn = {convex bodies in Rn}.
Kn is closed under addition and dilations: given K , L ∈ Kn andα, β ≥ 0,
αK + βL ∈ Kn.
The Brunn-Minkowski inequality holds in particular in Kn.
![Page 66: Around the Brunn-Minkowski inequality - UniFIweb.math.unifi.it/users/colesant/ricerca/Magdeburgo e Berlino 2015.pdf · Around the Brunn-Minkowski inequality Andrea Colesanti Technische](https://reader031.fdocuments.in/reader031/viewer/2022030820/5b33fad37f8b9a6b548ba38c/html5/thumbnails/66.jpg)
Convex bodies
From now on we will only consider a special type of compact sets:convex bodies.A convex body is a compact convex subset of Rn. We set
Kn = {convex bodies in Rn}.
Kn is closed under addition and dilations: given K , L ∈ Kn andα, β ≥ 0,
αK + βL ∈ Kn.
The Brunn-Minkowski inequality holds in particular in Kn.
![Page 67: Around the Brunn-Minkowski inequality - UniFIweb.math.unifi.it/users/colesant/ricerca/Magdeburgo e Berlino 2015.pdf · Around the Brunn-Minkowski inequality Andrea Colesanti Technische](https://reader031.fdocuments.in/reader031/viewer/2022030820/5b33fad37f8b9a6b548ba38c/html5/thumbnails/67.jpg)
From sets to functions:the support function of a convex body
I The support function hK of a convex body K is defined by:
hK : Sn−1 → R , hK (u) = sup{(u, v) |v ∈ K} .
hK (u) is the distance from the origin of the hyperplanesupporting K , with outer unit normal u.
I The passage to support functions preserves the linearstructure on Kn:
hαK+βL = αhK + βhL .
for every K , L ∈ Kn α, β ≥ 0.
![Page 68: Around the Brunn-Minkowski inequality - UniFIweb.math.unifi.it/users/colesant/ricerca/Magdeburgo e Berlino 2015.pdf · Around the Brunn-Minkowski inequality Andrea Colesanti Technische](https://reader031.fdocuments.in/reader031/viewer/2022030820/5b33fad37f8b9a6b548ba38c/html5/thumbnails/68.jpg)
From sets to functions:the support function of a convex body
I The support function hK of a convex body K is defined by:
hK : Sn−1 → R , hK (u) = sup{(u, v) |v ∈ K} .
hK (u) is the distance from the origin of the hyperplanesupporting K , with outer unit normal u.
I The passage to support functions preserves the linearstructure on Kn:
hαK+βL = αhK + βhL .
for every K , L ∈ Kn α, β ≥ 0.
![Page 69: Around the Brunn-Minkowski inequality - UniFIweb.math.unifi.it/users/colesant/ricerca/Magdeburgo e Berlino 2015.pdf · Around the Brunn-Minkowski inequality Andrea Colesanti Technische](https://reader031.fdocuments.in/reader031/viewer/2022030820/5b33fad37f8b9a6b548ba38c/html5/thumbnails/69.jpg)
From sets to functions:the support function of a convex body
I The support function hK of a convex body K is defined by:
hK : Sn−1 → R , hK (u) = sup{(u, v) |v ∈ K} .
hK (u) is the distance from the origin of the hyperplanesupporting K , with outer unit normal u.
I The passage to support functions preserves the linearstructure on Kn:
hαK+βL = αhK + βhL .
for every K , L ∈ Kn α, β ≥ 0.
![Page 70: Around the Brunn-Minkowski inequality - UniFIweb.math.unifi.it/users/colesant/ricerca/Magdeburgo e Berlino 2015.pdf · Around the Brunn-Minkowski inequality Andrea Colesanti Technische](https://reader031.fdocuments.in/reader031/viewer/2022030820/5b33fad37f8b9a6b548ba38c/html5/thumbnails/70.jpg)
Convex bodies of class C 2+
A convex body is said to be of class C 2+ if:
I ∂K ∈ C 2,
I the Gauss curvature is strictly positive on ∂K .
In terms of the support function h of K :
h ∈ C 2(Sn−1) , (hij + hδij) > 0 on Sn−1
(hij = second covariant derivatives of h on Sn−1, δij =Kronecker’s symbols).
C := {h ∈ C 2(Sn−1) : (hij + hδij) > 0 on Sn−1}= {support functions of C 2
+ convex bodies}.
![Page 71: Around the Brunn-Minkowski inequality - UniFIweb.math.unifi.it/users/colesant/ricerca/Magdeburgo e Berlino 2015.pdf · Around the Brunn-Minkowski inequality Andrea Colesanti Technische](https://reader031.fdocuments.in/reader031/viewer/2022030820/5b33fad37f8b9a6b548ba38c/html5/thumbnails/71.jpg)
Convex bodies of class C 2+
A convex body is said to be of class C 2+ if:
I ∂K ∈ C 2,
I the Gauss curvature is strictly positive on ∂K .
In terms of the support function h of K :
h ∈ C 2(Sn−1) , (hij + hδij) > 0 on Sn−1
(hij = second covariant derivatives of h on Sn−1, δij =Kronecker’s symbols).
C := {h ∈ C 2(Sn−1) : (hij + hδij) > 0 on Sn−1}= {support functions of C 2
+ convex bodies}.
![Page 72: Around the Brunn-Minkowski inequality - UniFIweb.math.unifi.it/users/colesant/ricerca/Magdeburgo e Berlino 2015.pdf · Around the Brunn-Minkowski inequality Andrea Colesanti Technische](https://reader031.fdocuments.in/reader031/viewer/2022030820/5b33fad37f8b9a6b548ba38c/html5/thumbnails/72.jpg)
Convex bodies of class C 2+
A convex body is said to be of class C 2+ if:
I ∂K ∈ C 2,
I the Gauss curvature is strictly positive on ∂K .
In terms of the support function h of K :
h ∈ C 2(Sn−1) ,
(hij + hδij) > 0 on Sn−1
(hij = second covariant derivatives of h on Sn−1, δij =Kronecker’s symbols).
C := {h ∈ C 2(Sn−1) : (hij + hδij) > 0 on Sn−1}= {support functions of C 2
+ convex bodies}.
![Page 73: Around the Brunn-Minkowski inequality - UniFIweb.math.unifi.it/users/colesant/ricerca/Magdeburgo e Berlino 2015.pdf · Around the Brunn-Minkowski inequality Andrea Colesanti Technische](https://reader031.fdocuments.in/reader031/viewer/2022030820/5b33fad37f8b9a6b548ba38c/html5/thumbnails/73.jpg)
Convex bodies of class C 2+
A convex body is said to be of class C 2+ if:
I ∂K ∈ C 2,
I the Gauss curvature is strictly positive on ∂K .
In terms of the support function h of K :
h ∈ C 2(Sn−1) , (hij + hδij) > 0 on Sn−1
(hij = second covariant derivatives of h on Sn−1, δij =Kronecker’s symbols).
C := {h ∈ C 2(Sn−1) : (hij + hδij) > 0 on Sn−1}= {support functions of C 2
+ convex bodies}.
![Page 74: Around the Brunn-Minkowski inequality - UniFIweb.math.unifi.it/users/colesant/ricerca/Magdeburgo e Berlino 2015.pdf · Around the Brunn-Minkowski inequality Andrea Colesanti Technische](https://reader031.fdocuments.in/reader031/viewer/2022030820/5b33fad37f8b9a6b548ba38c/html5/thumbnails/74.jpg)
Convex bodies of class C 2+
A convex body is said to be of class C 2+ if:
I ∂K ∈ C 2,
I the Gauss curvature is strictly positive on ∂K .
In terms of the support function h of K :
h ∈ C 2(Sn−1) , (hij + hδij) > 0 on Sn−1
(hij = second covariant derivatives of h on Sn−1, δij =Kronecker’s symbols).
C := {h ∈ C 2(Sn−1) : (hij + hδij) > 0 on Sn−1}
= {support functions of C 2+ convex bodies}.
![Page 75: Around the Brunn-Minkowski inequality - UniFIweb.math.unifi.it/users/colesant/ricerca/Magdeburgo e Berlino 2015.pdf · Around the Brunn-Minkowski inequality Andrea Colesanti Technische](https://reader031.fdocuments.in/reader031/viewer/2022030820/5b33fad37f8b9a6b548ba38c/html5/thumbnails/75.jpg)
Convex bodies of class C 2+
A convex body is said to be of class C 2+ if:
I ∂K ∈ C 2,
I the Gauss curvature is strictly positive on ∂K .
In terms of the support function h of K :
h ∈ C 2(Sn−1) , (hij + hδij) > 0 on Sn−1
(hij = second covariant derivatives of h on Sn−1, δij =Kronecker’s symbols).
C := {h ∈ C 2(Sn−1) : (hij + hδij) > 0 on Sn−1}= {support functions of C 2
+ convex bodies}.
![Page 76: Around the Brunn-Minkowski inequality - UniFIweb.math.unifi.it/users/colesant/ricerca/Magdeburgo e Berlino 2015.pdf · Around the Brunn-Minkowski inequality Andrea Colesanti Technische](https://reader031.fdocuments.in/reader031/viewer/2022030820/5b33fad37f8b9a6b548ba38c/html5/thumbnails/76.jpg)
A representation formula for the volume
If K is of class C 2+ and h is its support function, then
Vn(K ) =1
n
∫Sn−1
h det(hij + hδij) dHn−1.
Now we define a functional F : C → R+ as
F(h) =
[1
n
∫Sn−1
h det(hij + hδij) dHn−1]1/n
= Vn(K )1/n.
By the Brunn-Minkowski inequality,
F is concave in C.
![Page 77: Around the Brunn-Minkowski inequality - UniFIweb.math.unifi.it/users/colesant/ricerca/Magdeburgo e Berlino 2015.pdf · Around the Brunn-Minkowski inequality Andrea Colesanti Technische](https://reader031.fdocuments.in/reader031/viewer/2022030820/5b33fad37f8b9a6b548ba38c/html5/thumbnails/77.jpg)
A representation formula for the volume
If K is of class C 2+ and h is its support function, then
Vn(K ) =1
n
∫Sn−1
h det(hij + hδij) dHn−1.
Now we define a functional F : C → R+ as
F(h) =
[1
n
∫Sn−1
h det(hij + hδij) dHn−1]1/n
= Vn(K )1/n.
By the Brunn-Minkowski inequality,
F is concave in C.
![Page 78: Around the Brunn-Minkowski inequality - UniFIweb.math.unifi.it/users/colesant/ricerca/Magdeburgo e Berlino 2015.pdf · Around the Brunn-Minkowski inequality Andrea Colesanti Technische](https://reader031.fdocuments.in/reader031/viewer/2022030820/5b33fad37f8b9a6b548ba38c/html5/thumbnails/78.jpg)
A representation formula for the volume
If K is of class C 2+ and h is its support function, then
Vn(K ) =1
n
∫Sn−1
h det(hij + hδij) dHn−1.
Now we define a functional F : C → R+ as
F(h) =
[1
n
∫Sn−1
h det(hij + hδij) dHn−1]1/n
= Vn(K )1/n.
By the Brunn-Minkowski inequality,
F is concave in C.
![Page 79: Around the Brunn-Minkowski inequality - UniFIweb.math.unifi.it/users/colesant/ricerca/Magdeburgo e Berlino 2015.pdf · Around the Brunn-Minkowski inequality Andrea Colesanti Technische](https://reader031.fdocuments.in/reader031/viewer/2022030820/5b33fad37f8b9a6b548ba38c/html5/thumbnails/79.jpg)
A representation formula for the volume
If K is of class C 2+ and h is its support function, then
Vn(K ) =1
n
∫Sn−1
h det(hij + hδij) dHn−1.
Now we define a functional F : C → R+ as
F(h) =
[1
n
∫Sn−1
h det(hij + hδij) dHn−1]1/n
= Vn(K )1/n.
By the Brunn-Minkowski inequality,
F is concave in C.
![Page 80: Around the Brunn-Minkowski inequality - UniFIweb.math.unifi.it/users/colesant/ricerca/Magdeburgo e Berlino 2015.pdf · Around the Brunn-Minkowski inequality Andrea Colesanti Technische](https://reader031.fdocuments.in/reader031/viewer/2022030820/5b33fad37f8b9a6b548ba38c/html5/thumbnails/80.jpg)
The second variation of F.
F(h) =
[1
n
∫Sn−1
h det(hij + hδij) dHn−1]1/n
.
For every fixed h, D2F(h) is a bilinear symmetric form acting ontest functions φ ∈ C∞(Sn−1):
(D2F(h)φ, φ) =d2
ds2F(h + sφ)|s=0.
The condition(D2F(h)φ, φ) ≤ 0
turns out to be equivalent to a weighted Poincare inequality:∫Sn−1
trace(cij)φ2dHn−1 ≤
∫Sn−1
∑i ,j
cijφiφjdHn−1,
(cij) > 0, (cij) depends on h.
for every φ verifying a zero-mean condition. .
![Page 81: Around the Brunn-Minkowski inequality - UniFIweb.math.unifi.it/users/colesant/ricerca/Magdeburgo e Berlino 2015.pdf · Around the Brunn-Minkowski inequality Andrea Colesanti Technische](https://reader031.fdocuments.in/reader031/viewer/2022030820/5b33fad37f8b9a6b548ba38c/html5/thumbnails/81.jpg)
The second variation of F.
F(h) =
[1
n
∫Sn−1
h det(hij + hδij) dHn−1]1/n
.
For every fixed h, D2F(h) is a bilinear symmetric form acting ontest functions φ ∈ C∞(Sn−1):
(D2F(h)φ, φ) =d2
ds2F(h + sφ)|s=0.
The condition(D2F(h)φ, φ) ≤ 0
turns out to be equivalent to a weighted Poincare inequality:∫Sn−1
trace(cij)φ2dHn−1 ≤
∫Sn−1
∑i ,j
cijφiφjdHn−1,
(cij) > 0, (cij) depends on h.
for every φ verifying a zero-mean condition. .
![Page 82: Around the Brunn-Minkowski inequality - UniFIweb.math.unifi.it/users/colesant/ricerca/Magdeburgo e Berlino 2015.pdf · Around the Brunn-Minkowski inequality Andrea Colesanti Technische](https://reader031.fdocuments.in/reader031/viewer/2022030820/5b33fad37f8b9a6b548ba38c/html5/thumbnails/82.jpg)
The second variation of F.
F(h) =
[1
n
∫Sn−1
h det(hij + hδij) dHn−1]1/n
.
For every fixed h, D2F(h) is a bilinear symmetric form acting ontest functions φ ∈ C∞(Sn−1):
(D2F(h)φ, φ) =d2
ds2F(h + sφ)|s=0.
The condition(D2F(h)φ, φ) ≤ 0
turns out to be equivalent to a weighted Poincare inequality:∫Sn−1
trace(cij)φ2dHn−1 ≤
∫Sn−1
∑i ,j
cijφiφjdHn−1,
(cij) > 0, (cij) depends on h.
for every φ verifying a zero-mean condition. .
![Page 83: Around the Brunn-Minkowski inequality - UniFIweb.math.unifi.it/users/colesant/ricerca/Magdeburgo e Berlino 2015.pdf · Around the Brunn-Minkowski inequality Andrea Colesanti Technische](https://reader031.fdocuments.in/reader031/viewer/2022030820/5b33fad37f8b9a6b548ba38c/html5/thumbnails/83.jpg)
The second variation of F.
F(h) =
[1
n
∫Sn−1
h det(hij + hδij) dHn−1]1/n
.
For every fixed h, D2F(h) is a bilinear symmetric form acting ontest functions φ ∈ C∞(Sn−1):
(D2F(h)φ, φ)
=d2
ds2F(h + sφ)|s=0.
The condition(D2F(h)φ, φ) ≤ 0
turns out to be equivalent to a weighted Poincare inequality:∫Sn−1
trace(cij)φ2dHn−1 ≤
∫Sn−1
∑i ,j
cijφiφjdHn−1,
(cij) > 0, (cij) depends on h.
for every φ verifying a zero-mean condition. .
![Page 84: Around the Brunn-Minkowski inequality - UniFIweb.math.unifi.it/users/colesant/ricerca/Magdeburgo e Berlino 2015.pdf · Around the Brunn-Minkowski inequality Andrea Colesanti Technische](https://reader031.fdocuments.in/reader031/viewer/2022030820/5b33fad37f8b9a6b548ba38c/html5/thumbnails/84.jpg)
The second variation of F.
F(h) =
[1
n
∫Sn−1
h det(hij + hδij) dHn−1]1/n
.
For every fixed h, D2F(h) is a bilinear symmetric form acting ontest functions φ ∈ C∞(Sn−1):
(D2F(h)φ, φ) =d2
ds2F(h + sφ)|s=0.
The condition(D2F(h)φ, φ) ≤ 0
turns out to be equivalent to a weighted Poincare inequality:∫Sn−1
trace(cij)φ2dHn−1 ≤
∫Sn−1
∑i ,j
cijφiφjdHn−1,
(cij) > 0, (cij) depends on h.
for every φ verifying a zero-mean condition. .
![Page 85: Around the Brunn-Minkowski inequality - UniFIweb.math.unifi.it/users/colesant/ricerca/Magdeburgo e Berlino 2015.pdf · Around the Brunn-Minkowski inequality Andrea Colesanti Technische](https://reader031.fdocuments.in/reader031/viewer/2022030820/5b33fad37f8b9a6b548ba38c/html5/thumbnails/85.jpg)
The second variation of F.
F(h) =
[1
n
∫Sn−1
h det(hij + hδij) dHn−1]1/n
.
For every fixed h, D2F(h) is a bilinear symmetric form acting ontest functions φ ∈ C∞(Sn−1):
(D2F(h)φ, φ) =d2
ds2F(h + sφ)|s=0.
The condition(D2F(h)φ, φ) ≤ 0
turns out to be equivalent to a weighted Poincare inequality:∫Sn−1
trace(cij)φ2dHn−1 ≤
∫Sn−1
∑i ,j
cijφiφjdHn−1,
(cij) > 0, (cij) depends on h.
for every φ verifying a zero-mean condition. .
![Page 86: Around the Brunn-Minkowski inequality - UniFIweb.math.unifi.it/users/colesant/ricerca/Magdeburgo e Berlino 2015.pdf · Around the Brunn-Minkowski inequality Andrea Colesanti Technische](https://reader031.fdocuments.in/reader031/viewer/2022030820/5b33fad37f8b9a6b548ba38c/html5/thumbnails/86.jpg)
The second variation of F.
F(h) =
[1
n
∫Sn−1
h det(hij + hδij) dHn−1]1/n
.
For every fixed h, D2F(h) is a bilinear symmetric form acting ontest functions φ ∈ C∞(Sn−1):
(D2F(h)φ, φ) =d2
ds2F(h + sφ)|s=0.
The condition(D2F(h)φ, φ) ≤ 0
turns out to be equivalent to a weighted Poincare inequality:∫Sn−1
trace(cij)φ2dHn−1 ≤
∫Sn−1
∑i ,j
cijφiφjdHn−1,
(cij) > 0, (cij) depends on h.
for every φ verifying a zero-mean condition. .
![Page 87: Around the Brunn-Minkowski inequality - UniFIweb.math.unifi.it/users/colesant/ricerca/Magdeburgo e Berlino 2015.pdf · Around the Brunn-Minkowski inequality Andrea Colesanti Technische](https://reader031.fdocuments.in/reader031/viewer/2022030820/5b33fad37f8b9a6b548ba38c/html5/thumbnails/87.jpg)
The second variation of F.
F(h) =
[1
n
∫Sn−1
h det(hij + hδij) dHn−1]1/n
.
For every fixed h, D2F(h) is a bilinear symmetric form acting ontest functions φ ∈ C∞(Sn−1):
(D2F(h)φ, φ) =d2
ds2F(h + sφ)|s=0.
The condition(D2F(h)φ, φ) ≤ 0
turns out to be equivalent to a weighted Poincare inequality:∫Sn−1
trace(cij)φ2dHn−1 ≤
∫Sn−1
∑i ,j
cijφiφjdHn−1,
(cij) > 0, (cij) depends on h.
for every φ verifying a zero-mean condition. .
![Page 88: Around the Brunn-Minkowski inequality - UniFIweb.math.unifi.it/users/colesant/ricerca/Magdeburgo e Berlino 2015.pdf · Around the Brunn-Minkowski inequality Andrea Colesanti Technische](https://reader031.fdocuments.in/reader031/viewer/2022030820/5b33fad37f8b9a6b548ba38c/html5/thumbnails/88.jpg)
The second variation of F.
F(h) =
[1
n
∫Sn−1
h det(hij + hδij) dHn−1]1/n
.
For every fixed h, D2F(h) is a bilinear symmetric form acting ontest functions φ ∈ C∞(Sn−1):
(D2F(h)φ, φ) =d2
ds2F(h + sφ)|s=0.
The condition(D2F(h)φ, φ) ≤ 0
turns out to be equivalent to a weighted Poincare inequality:∫Sn−1
trace(cij)φ2dHn−1 ≤
∫Sn−1
∑i ,j
cijφiφjdHn−1,
(cij) > 0, (cij) depends on h.
for every φ verifying a zero-mean condition. .
![Page 89: Around the Brunn-Minkowski inequality - UniFIweb.math.unifi.it/users/colesant/ricerca/Magdeburgo e Berlino 2015.pdf · Around the Brunn-Minkowski inequality Andrea Colesanti Technische](https://reader031.fdocuments.in/reader031/viewer/2022030820/5b33fad37f8b9a6b548ba38c/html5/thumbnails/89.jpg)
A special case
If we choose h ≡ 1 (the support function of the unit ball of Rn),we obtain (cij) =identity matrix, and we recover∫
Sn−1
φ2dHn−1 ≤ 1
n − 1
∫Sn−1
|∇φ|2dHn−1,
for every φ ∈ C 1(Sn−1) s.t.∫Sn−1
φdHn−1 = 0.
This is the standard Poincare inequality (with best constant) onSn−1.(C. 2008; Saorın-Gomez, C. 2010).
![Page 90: Around the Brunn-Minkowski inequality - UniFIweb.math.unifi.it/users/colesant/ricerca/Magdeburgo e Berlino 2015.pdf · Around the Brunn-Minkowski inequality Andrea Colesanti Technische](https://reader031.fdocuments.in/reader031/viewer/2022030820/5b33fad37f8b9a6b548ba38c/html5/thumbnails/90.jpg)
A special case
If we choose h ≡ 1 (the support function of the unit ball of Rn),we obtain (cij) =identity matrix, and we recover∫
Sn−1
φ2dHn−1 ≤ 1
n − 1
∫Sn−1
|∇φ|2dHn−1,
for every φ ∈ C 1(Sn−1) s.t.∫Sn−1
φdHn−1 = 0.
This is the standard Poincare inequality (with best constant) onSn−1.(C. 2008; Saorın-Gomez, C. 2010).
![Page 91: Around the Brunn-Minkowski inequality - UniFIweb.math.unifi.it/users/colesant/ricerca/Magdeburgo e Berlino 2015.pdf · Around the Brunn-Minkowski inequality Andrea Colesanti Technische](https://reader031.fdocuments.in/reader031/viewer/2022030820/5b33fad37f8b9a6b548ba38c/html5/thumbnails/91.jpg)
A special case
If we choose h ≡ 1 (the support function of the unit ball of Rn),we obtain (cij) =identity matrix, and we recover∫
Sn−1
φ2dHn−1 ≤ 1
n − 1
∫Sn−1
|∇φ|2dHn−1,
for every φ ∈ C 1(Sn−1) s.t.∫Sn−1
φdHn−1 = 0.
This is the standard Poincare inequality (with best constant) onSn−1.
(C. 2008; Saorın-Gomez, C. 2010).
![Page 92: Around the Brunn-Minkowski inequality - UniFIweb.math.unifi.it/users/colesant/ricerca/Magdeburgo e Berlino 2015.pdf · Around the Brunn-Minkowski inequality Andrea Colesanti Technische](https://reader031.fdocuments.in/reader031/viewer/2022030820/5b33fad37f8b9a6b548ba38c/html5/thumbnails/92.jpg)
A special case
If we choose h ≡ 1 (the support function of the unit ball of Rn),we obtain (cij) =identity matrix, and we recover∫
Sn−1
φ2dHn−1 ≤ 1
n − 1
∫Sn−1
|∇φ|2dHn−1,
for every φ ∈ C 1(Sn−1) s.t.∫Sn−1
φdHn−1 = 0.
This is the standard Poincare inequality (with best constant) onSn−1.(C. 2008; Saorın-Gomez, C. 2010).
![Page 93: Around the Brunn-Minkowski inequality - UniFIweb.math.unifi.it/users/colesant/ricerca/Magdeburgo e Berlino 2015.pdf · Around the Brunn-Minkowski inequality Andrea Colesanti Technische](https://reader031.fdocuments.in/reader031/viewer/2022030820/5b33fad37f8b9a6b548ba38c/html5/thumbnails/93.jpg)
Inequalities of Brunn-Minkowski type
Let G : Kn → R be s.t.:
I G(K ) ≥ 0 for every K ∈ Kn;
I G is α-homogeneous (α 6= 0):
G(tK ) = tα G(K ), ∀ t ≥ 0, K ∈ Kn.
—————–
We say that G verifies a Brunn-Minkowski type inequality if forevery K , L ∈ Kn, and for every λ ∈ [0, 1]:
G((1− λ)K + λL)1/α ≥ (1− λ)G(K )1/α + λG(L)1/α.
![Page 94: Around the Brunn-Minkowski inequality - UniFIweb.math.unifi.it/users/colesant/ricerca/Magdeburgo e Berlino 2015.pdf · Around the Brunn-Minkowski inequality Andrea Colesanti Technische](https://reader031.fdocuments.in/reader031/viewer/2022030820/5b33fad37f8b9a6b548ba38c/html5/thumbnails/94.jpg)
Inequalities of Brunn-Minkowski type
Let G : Kn → R be s.t.:
I G(K ) ≥ 0 for every K ∈ Kn;
I G is α-homogeneous (α 6= 0):
G(tK ) = tα G(K ), ∀ t ≥ 0, K ∈ Kn.
—————–
We say that G verifies a Brunn-Minkowski type inequality if forevery K , L ∈ Kn, and for every λ ∈ [0, 1]:
G((1− λ)K + λL)1/α ≥ (1− λ)G(K )1/α + λG(L)1/α.
![Page 95: Around the Brunn-Minkowski inequality - UniFIweb.math.unifi.it/users/colesant/ricerca/Magdeburgo e Berlino 2015.pdf · Around the Brunn-Minkowski inequality Andrea Colesanti Technische](https://reader031.fdocuments.in/reader031/viewer/2022030820/5b33fad37f8b9a6b548ba38c/html5/thumbnails/95.jpg)
Inequalities of Brunn-Minkowski type
Let G : Kn → R be s.t.:
I G(K ) ≥ 0 for every K ∈ Kn;
I G is α-homogeneous (α 6= 0):
G(tK ) = tα G(K ), ∀ t ≥ 0, K ∈ Kn.
—————–
We say that G verifies a Brunn-Minkowski type inequality if forevery K , L ∈ Kn, and for every λ ∈ [0, 1]:
G((1− λ)K + λL)1/α ≥ (1− λ)G(K )1/α + λG(L)1/α.
![Page 96: Around the Brunn-Minkowski inequality - UniFIweb.math.unifi.it/users/colesant/ricerca/Magdeburgo e Berlino 2015.pdf · Around the Brunn-Minkowski inequality Andrea Colesanti Technische](https://reader031.fdocuments.in/reader031/viewer/2022030820/5b33fad37f8b9a6b548ba38c/html5/thumbnails/96.jpg)
Examples
The following functionals verify a Brunn-Minkowski type inequality.
I Volume.
I Perimeter.
I Other functionals in convex geometry (intrinsic volumes,mixed volumes, 2-dim. affine surface area...).
I Principal frequency (= first Dirichlet eigenvalue of the Laplaceoperator) (Brascamp and Lieb; Borell).
I Electrostatic capacity (Borell; Caffarelli, Jerison and Lieb).
I Many other examples coming from the world of calculus ofvariations and elliptic PDE’s (torsional rigidity, p-capacity, ...).
![Page 97: Around the Brunn-Minkowski inequality - UniFIweb.math.unifi.it/users/colesant/ricerca/Magdeburgo e Berlino 2015.pdf · Around the Brunn-Minkowski inequality Andrea Colesanti Technische](https://reader031.fdocuments.in/reader031/viewer/2022030820/5b33fad37f8b9a6b548ba38c/html5/thumbnails/97.jpg)
Examples
The following functionals verify a Brunn-Minkowski type inequality.
I Volume.
I Perimeter.
I Other functionals in convex geometry (intrinsic volumes,mixed volumes, 2-dim. affine surface area...).
I Principal frequency (= first Dirichlet eigenvalue of the Laplaceoperator) (Brascamp and Lieb; Borell).
I Electrostatic capacity (Borell; Caffarelli, Jerison and Lieb).
I Many other examples coming from the world of calculus ofvariations and elliptic PDE’s (torsional rigidity, p-capacity, ...).
![Page 98: Around the Brunn-Minkowski inequality - UniFIweb.math.unifi.it/users/colesant/ricerca/Magdeburgo e Berlino 2015.pdf · Around the Brunn-Minkowski inequality Andrea Colesanti Technische](https://reader031.fdocuments.in/reader031/viewer/2022030820/5b33fad37f8b9a6b548ba38c/html5/thumbnails/98.jpg)
Examples
The following functionals verify a Brunn-Minkowski type inequality.
I Volume.
I Perimeter.
I Other functionals in convex geometry (intrinsic volumes,mixed volumes, 2-dim. affine surface area...).
I Principal frequency (= first Dirichlet eigenvalue of the Laplaceoperator) (Brascamp and Lieb; Borell).
I Electrostatic capacity (Borell; Caffarelli, Jerison and Lieb).
I Many other examples coming from the world of calculus ofvariations and elliptic PDE’s (torsional rigidity, p-capacity, ...).
![Page 99: Around the Brunn-Minkowski inequality - UniFIweb.math.unifi.it/users/colesant/ricerca/Magdeburgo e Berlino 2015.pdf · Around the Brunn-Minkowski inequality Andrea Colesanti Technische](https://reader031.fdocuments.in/reader031/viewer/2022030820/5b33fad37f8b9a6b548ba38c/html5/thumbnails/99.jpg)
Examples
The following functionals verify a Brunn-Minkowski type inequality.
I Volume.
I Perimeter.
I Other functionals in convex geometry (intrinsic volumes,mixed volumes, 2-dim. affine surface area...).
I Principal frequency (= first Dirichlet eigenvalue of the Laplaceoperator) (Brascamp and Lieb; Borell).
I Electrostatic capacity (Borell; Caffarelli, Jerison and Lieb).
I Many other examples coming from the world of calculus ofvariations and elliptic PDE’s (torsional rigidity, p-capacity, ...).
![Page 100: Around the Brunn-Minkowski inequality - UniFIweb.math.unifi.it/users/colesant/ricerca/Magdeburgo e Berlino 2015.pdf · Around the Brunn-Minkowski inequality Andrea Colesanti Technische](https://reader031.fdocuments.in/reader031/viewer/2022030820/5b33fad37f8b9a6b548ba38c/html5/thumbnails/100.jpg)
Examples
The following functionals verify a Brunn-Minkowski type inequality.
I Volume.
I Perimeter.
I Other functionals in convex geometry (intrinsic volumes,mixed volumes, 2-dim. affine surface area...).
I Principal frequency (= first Dirichlet eigenvalue of the Laplaceoperator) (Brascamp and Lieb; Borell).
I Electrostatic capacity (Borell; Caffarelli, Jerison and Lieb).
I Many other examples coming from the world of calculus ofvariations and elliptic PDE’s (torsional rigidity, p-capacity, ...).
![Page 101: Around the Brunn-Minkowski inequality - UniFIweb.math.unifi.it/users/colesant/ricerca/Magdeburgo e Berlino 2015.pdf · Around the Brunn-Minkowski inequality Andrea Colesanti Technische](https://reader031.fdocuments.in/reader031/viewer/2022030820/5b33fad37f8b9a6b548ba38c/html5/thumbnails/101.jpg)
Examples
The following functionals verify a Brunn-Minkowski type inequality.
I Volume.
I Perimeter.
I Other functionals in convex geometry (intrinsic volumes,mixed volumes, 2-dim. affine surface area...).
I Principal frequency (= first Dirichlet eigenvalue of the Laplaceoperator) (Brascamp and Lieb; Borell).
I Electrostatic capacity (Borell; Caffarelli, Jerison and Lieb).
I Many other examples coming from the world of calculus ofvariations and elliptic PDE’s (torsional rigidity, p-capacity, ...).
![Page 102: Around the Brunn-Minkowski inequality - UniFIweb.math.unifi.it/users/colesant/ricerca/Magdeburgo e Berlino 2015.pdf · Around the Brunn-Minkowski inequality Andrea Colesanti Technische](https://reader031.fdocuments.in/reader031/viewer/2022030820/5b33fad37f8b9a6b548ba38c/html5/thumbnails/102.jpg)
Examples
The following functionals verify a Brunn-Minkowski type inequality.
I Volume.
I Perimeter.
I Other functionals in convex geometry (intrinsic volumes,mixed volumes, 2-dim. affine surface area...).
I Principal frequency (= first Dirichlet eigenvalue of the Laplaceoperator) (Brascamp and Lieb; Borell).
I Electrostatic capacity (Borell; Caffarelli, Jerison and Lieb).
I Many other examples coming from the world of calculus ofvariations and elliptic PDE’s (torsional rigidity, p-capacity, ...).
![Page 103: Around the Brunn-Minkowski inequality - UniFIweb.math.unifi.it/users/colesant/ricerca/Magdeburgo e Berlino 2015.pdf · Around the Brunn-Minkowski inequality Andrea Colesanti Technische](https://reader031.fdocuments.in/reader031/viewer/2022030820/5b33fad37f8b9a6b548ba38c/html5/thumbnails/103.jpg)
Examples
The following functionals verify a Brunn-Minkowski type inequality.
I Volume.
I Perimeter.
I Other functionals in convex geometry (intrinsic volumes,mixed volumes, 2-dim. affine surface area...).
I Principal frequency (= first Dirichlet eigenvalue of the Laplaceoperator) (Brascamp and Lieb; Borell).
I Electrostatic capacity (Borell; Caffarelli, Jerison and Lieb).
I Many other examples coming from the world of calculus ofvariations and elliptic PDE’s (torsional rigidity, p-capacity, ...).
![Page 104: Around the Brunn-Minkowski inequality - UniFIweb.math.unifi.it/users/colesant/ricerca/Magdeburgo e Berlino 2015.pdf · Around the Brunn-Minkowski inequality Andrea Colesanti Technische](https://reader031.fdocuments.in/reader031/viewer/2022030820/5b33fad37f8b9a6b548ba38c/html5/thumbnails/104.jpg)
Other examples
Some well-known functional not obeying a Brunn-Minkowskiinequality.
I The diameter.
I The affine surface area in dimension n ≥ 3.
I The first Neumann eigenvalue of the Laplace operator.
![Page 105: Around the Brunn-Minkowski inequality - UniFIweb.math.unifi.it/users/colesant/ricerca/Magdeburgo e Berlino 2015.pdf · Around the Brunn-Minkowski inequality Andrea Colesanti Technische](https://reader031.fdocuments.in/reader031/viewer/2022030820/5b33fad37f8b9a6b548ba38c/html5/thumbnails/105.jpg)
Other examples
Some well-known functional not obeying a Brunn-Minkowskiinequality.
I The diameter.
I The affine surface area in dimension n ≥ 3.
I The first Neumann eigenvalue of the Laplace operator.
![Page 106: Around the Brunn-Minkowski inequality - UniFIweb.math.unifi.it/users/colesant/ricerca/Magdeburgo e Berlino 2015.pdf · Around the Brunn-Minkowski inequality Andrea Colesanti Technische](https://reader031.fdocuments.in/reader031/viewer/2022030820/5b33fad37f8b9a6b548ba38c/html5/thumbnails/106.jpg)
Other examples
Some well-known functional not obeying a Brunn-Minkowskiinequality.
I The diameter.
I The affine surface area in dimension n ≥ 3.
I The first Neumann eigenvalue of the Laplace operator.
![Page 107: Around the Brunn-Minkowski inequality - UniFIweb.math.unifi.it/users/colesant/ricerca/Magdeburgo e Berlino 2015.pdf · Around the Brunn-Minkowski inequality Andrea Colesanti Technische](https://reader031.fdocuments.in/reader031/viewer/2022030820/5b33fad37f8b9a6b548ba38c/html5/thumbnails/107.jpg)
Other examples
Some well-known functional not obeying a Brunn-Minkowskiinequality.
I The diameter.
I The affine surface area in dimension n ≥ 3.
I The first Neumann eigenvalue of the Laplace operator.
![Page 108: Around the Brunn-Minkowski inequality - UniFIweb.math.unifi.it/users/colesant/ricerca/Magdeburgo e Berlino 2015.pdf · Around the Brunn-Minkowski inequality Andrea Colesanti Technische](https://reader031.fdocuments.in/reader031/viewer/2022030820/5b33fad37f8b9a6b548ba38c/html5/thumbnails/108.jpg)
Other examples
Some well-known functional not obeying a Brunn-Minkowskiinequality.
I The diameter.
I The affine surface area in dimension n ≥ 3.
I The first Neumann eigenvalue of the Laplace operator.
![Page 109: Around the Brunn-Minkowski inequality - UniFIweb.math.unifi.it/users/colesant/ricerca/Magdeburgo e Berlino 2015.pdf · Around the Brunn-Minkowski inequality Andrea Colesanti Technische](https://reader031.fdocuments.in/reader031/viewer/2022030820/5b33fad37f8b9a6b548ba38c/html5/thumbnails/109.jpg)
Hints
Is there some general phenomenon behind these examples?Difficult (pointless?) to say.
Maybe simpler: understand the relation between theBrunn-Minkowski inequality and other basic features, such as:
I monotonicity;
I continuity;
I rigid motion invariance;
I additivity (or valuation property):
G(K ∪ L) = G(K ) + G(L)− G(K ∩ L),
for every K , L ∈ Kn such that
K ∪ L ∈ Kn.
![Page 110: Around the Brunn-Minkowski inequality - UniFIweb.math.unifi.it/users/colesant/ricerca/Magdeburgo e Berlino 2015.pdf · Around the Brunn-Minkowski inequality Andrea Colesanti Technische](https://reader031.fdocuments.in/reader031/viewer/2022030820/5b33fad37f8b9a6b548ba38c/html5/thumbnails/110.jpg)
Hints
Is there some general phenomenon behind these examples?
Difficult (pointless?) to say.
Maybe simpler: understand the relation between theBrunn-Minkowski inequality and other basic features, such as:
I monotonicity;
I continuity;
I rigid motion invariance;
I additivity (or valuation property):
G(K ∪ L) = G(K ) + G(L)− G(K ∩ L),
for every K , L ∈ Kn such that
K ∪ L ∈ Kn.
![Page 111: Around the Brunn-Minkowski inequality - UniFIweb.math.unifi.it/users/colesant/ricerca/Magdeburgo e Berlino 2015.pdf · Around the Brunn-Minkowski inequality Andrea Colesanti Technische](https://reader031.fdocuments.in/reader031/viewer/2022030820/5b33fad37f8b9a6b548ba38c/html5/thumbnails/111.jpg)
Hints
Is there some general phenomenon behind these examples?Difficult (pointless?) to say.
Maybe simpler: understand the relation between theBrunn-Minkowski inequality and other basic features, such as:
I monotonicity;
I continuity;
I rigid motion invariance;
I additivity (or valuation property):
G(K ∪ L) = G(K ) + G(L)− G(K ∩ L),
for every K , L ∈ Kn such that
K ∪ L ∈ Kn.
![Page 112: Around the Brunn-Minkowski inequality - UniFIweb.math.unifi.it/users/colesant/ricerca/Magdeburgo e Berlino 2015.pdf · Around the Brunn-Minkowski inequality Andrea Colesanti Technische](https://reader031.fdocuments.in/reader031/viewer/2022030820/5b33fad37f8b9a6b548ba38c/html5/thumbnails/112.jpg)
Hints
Is there some general phenomenon behind these examples?Difficult (pointless?) to say.
Maybe simpler: understand the relation between theBrunn-Minkowski inequality and other basic features, such as:
I monotonicity;
I continuity;
I rigid motion invariance;
I additivity (or valuation property):
G(K ∪ L) = G(K ) + G(L)− G(K ∩ L),
for every K , L ∈ Kn such that
K ∪ L ∈ Kn.
![Page 113: Around the Brunn-Minkowski inequality - UniFIweb.math.unifi.it/users/colesant/ricerca/Magdeburgo e Berlino 2015.pdf · Around the Brunn-Minkowski inequality Andrea Colesanti Technische](https://reader031.fdocuments.in/reader031/viewer/2022030820/5b33fad37f8b9a6b548ba38c/html5/thumbnails/113.jpg)
Hints
Is there some general phenomenon behind these examples?Difficult (pointless?) to say.
Maybe simpler: understand the relation between theBrunn-Minkowski inequality and other basic features, such as:
I monotonicity;
I continuity;
I rigid motion invariance;
I additivity (or valuation property):
G(K ∪ L) = G(K ) + G(L)− G(K ∩ L),
for every K , L ∈ Kn such that
K ∪ L ∈ Kn.
![Page 114: Around the Brunn-Minkowski inequality - UniFIweb.math.unifi.it/users/colesant/ricerca/Magdeburgo e Berlino 2015.pdf · Around the Brunn-Minkowski inequality Andrea Colesanti Technische](https://reader031.fdocuments.in/reader031/viewer/2022030820/5b33fad37f8b9a6b548ba38c/html5/thumbnails/114.jpg)
Hints
Is there some general phenomenon behind these examples?Difficult (pointless?) to say.
Maybe simpler: understand the relation between theBrunn-Minkowski inequality and other basic features, such as:
I monotonicity;
I continuity;
I rigid motion invariance;
I additivity (or valuation property):
G(K ∪ L) = G(K ) + G(L)− G(K ∩ L),
for every K , L ∈ Kn such that
K ∪ L ∈ Kn.
![Page 115: Around the Brunn-Minkowski inequality - UniFIweb.math.unifi.it/users/colesant/ricerca/Magdeburgo e Berlino 2015.pdf · Around the Brunn-Minkowski inequality Andrea Colesanti Technische](https://reader031.fdocuments.in/reader031/viewer/2022030820/5b33fad37f8b9a6b548ba38c/html5/thumbnails/115.jpg)
Hints
Is there some general phenomenon behind these examples?Difficult (pointless?) to say.
Maybe simpler: understand the relation between theBrunn-Minkowski inequality and other basic features, such as:
I monotonicity;
I continuity;
I rigid motion invariance;
I additivity (or valuation property):
G(K ∪ L) = G(K ) + G(L)− G(K ∩ L),
for every K , L ∈ Kn such that
K ∪ L ∈ Kn.
![Page 116: Around the Brunn-Minkowski inequality - UniFIweb.math.unifi.it/users/colesant/ricerca/Magdeburgo e Berlino 2015.pdf · Around the Brunn-Minkowski inequality Andrea Colesanti Technische](https://reader031.fdocuments.in/reader031/viewer/2022030820/5b33fad37f8b9a6b548ba38c/html5/thumbnails/116.jpg)
Hints
Is there some general phenomenon behind these examples?Difficult (pointless?) to say.
Maybe simpler: understand the relation between theBrunn-Minkowski inequality and other basic features, such as:
I monotonicity;
I continuity;
I rigid motion invariance;
I additivity (or valuation property):
G(K ∪ L) = G(K ) + G(L)− G(K ∩ L),
for every K , L ∈ Kn such that
K ∪ L ∈ Kn.
![Page 117: Around the Brunn-Minkowski inequality - UniFIweb.math.unifi.it/users/colesant/ricerca/Magdeburgo e Berlino 2015.pdf · Around the Brunn-Minkowski inequality Andrea Colesanti Technische](https://reader031.fdocuments.in/reader031/viewer/2022030820/5b33fad37f8b9a6b548ba38c/html5/thumbnails/117.jpg)
A result in this direction
Thm. (Hug, Saorın-Gomez, C., 2012). Let G : Kn → R be:additive, rigid motion invariant, continuous, (n − 1)-homogeneous,and assume that it verifies a Brunn-Minkowski type inequality.Then G is a mixed volume, and in particular is monotone.
![Page 118: Around the Brunn-Minkowski inequality - UniFIweb.math.unifi.it/users/colesant/ricerca/Magdeburgo e Berlino 2015.pdf · Around the Brunn-Minkowski inequality Andrea Colesanti Technische](https://reader031.fdocuments.in/reader031/viewer/2022030820/5b33fad37f8b9a6b548ba38c/html5/thumbnails/118.jpg)
A result in this direction
Thm. (Hug, Saorın-Gomez, C., 2012). Let G : Kn → R be:additive, rigid motion invariant, continuous, (n − 1)-homogeneous,and assume that it verifies a Brunn-Minkowski type inequality.
Then G is a mixed volume, and in particular is monotone.
![Page 119: Around the Brunn-Minkowski inequality - UniFIweb.math.unifi.it/users/colesant/ricerca/Magdeburgo e Berlino 2015.pdf · Around the Brunn-Minkowski inequality Andrea Colesanti Technische](https://reader031.fdocuments.in/reader031/viewer/2022030820/5b33fad37f8b9a6b548ba38c/html5/thumbnails/119.jpg)
A result in this direction
Thm. (Hug, Saorın-Gomez, C., 2012). Let G : Kn → R be:additive, rigid motion invariant, continuous, (n − 1)-homogeneous,and assume that it verifies a Brunn-Minkowski type inequality.Then G is a mixed volume, and in particular is monotone.