Probability and Statistics Cheat Sheet by Matthias Vallentin
Mathieu Dutour Sikiric, Konstantin Rybnikov, Achill Schuermann and Frank Vallentin- Covering maxima...
Transcript of Mathieu Dutour Sikiric, Konstantin Rybnikov, Achill Schuermann and Frank Vallentin- Covering maxima...
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8/3/2019 Mathieu Dutour Sikiric, Konstantin Rybnikov, Achill Schuermann and Frank Vallentin- Covering maxima and Delaunay polytopes in lattices
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Covering maxima and Delaunay polytopes in
lattices
Mathieu Dutour Sikiric
Rudjer Boskovic Institute,Croatia
Konstantin Rybnikov
Massachussets, Lowell, USA
Achill Schuermann
TU Delft, Netherland
Frank Vallentin
TU Delft, Netherland
January 16, 2010
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I. Quadratic functions
and the Erdahl cone
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The Erdahl cone
Denote by E2(n) the vector space of degree 2 polynomial
functions onRn
. We write f E2(n) in the form
f(x) = af + bf x + Qf[x]
with af R, bf Rn and Qf a n n symmetric matrix
The Erdahl cone is defined as
Erdahl(n) = {f E2(n) such that f(x) 0 for x Zn}
It is a convex cone, which is non-polyhedral since defined by
an infinity of inequalities. The group acting on Erdahl(n) is AGLn(Z), i.e. the group of
affine integral transformations
x b+ xP for b Zn and P GLn(Z)
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Scalar product
Definition: Iff, g
E
2(n
), then:
f, g = afag + bf, bg + Qf,Qg
Definition: For v Zn, define evv(x) = (1 + v x)2.
We have f, evv = f(v)
Thus finding the rays of Erdahl(n) is a dual descriptionproblem with an infinity of inequalities and infinite groupacting on it.
Definition: We also define
Erdahl>0(n) = {f Erdahl(n) : Qf positive definite}
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Empty sphere and Delaunay polytopes
Definition: A sphere S(c, r) of center c and radius r in an
n-dimensional lattice L is said to be an empty sphere if:(i) v c r for all v L,
(ii) the set S(c, r) L contains n + 1 affinely independent points.
Definition: A Delaunay polytope P in a lattice L is apolytope, whose vertex-set is L S(c, r).
c
r
Delaunay polytopes define a tesselation of the Euclideanspace.
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Relation with Delaunay polytope
If D is a Delaunay polytope of a lattice L of empty sphereS(c, r) then for every basis v1, . . . , vn of L we define the
function
fD : Zn R
x = (x1, . . . , xn) n
i=1 xivi c2 r2
Clearly fD Erdahl>0(n). On the other hand if f Erdahl(n) then there exist a lattice
L of dimension k n, a Delaunay polytope D of L, afunction AGLn(Z) such that
f (x1, . . . , xn) = fD(x1, . . . , xk)
Thus the faces of Erdahl(n) correspond to the Delaunaypolytope of dimension at most n.
The rank of a Delaunay polytope is the dimension of the face
it defines in Erdahl(n).
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Perfect quadratic function
Definition: If f Erdahl(n) then
Z(f) = {v Zn
: f(v) = 0}
andDom f =
vZ(f)
R+evv
We have f,Dom f = 0. Definition: f Erdahl(n) is perfect if Dom f is of dimension
n+22
1.
This is equivalent to say that f defines an extreme ray in
Erdahl(n). The interval [0, 1] is a perfect Delaunay polytope and for every
n 6 there exist an n-dimensional perfect Delaunay polytope.
Geometrically, this means that the only affine transformationsthat preserve the Delaunay property are isometries and
homotheties.
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Eutacticity
If f Erdahl>0(n) then define f and cf such that
f(x) = Qf[x cf] f
Then define
uf(x) = (1 + cf x)2 +
f
nQ1f [x]
Definition: f Erdahl>0(n) is eutactic if uf relint(Dom f)and weakly eutactic if uf Dom f.
Definition: Take a Delaunay polytope P for a quadratic formQ of center cP and square radius P. P is called eutactic ifthere are v > 0 so that
1 =
vvertPv,
0 =
vvertPv(v cP),
Pn
Q1 = vvertP
v(v cP)(v cP)t.
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Strongly perfect Delaunay polytopes
Definition: We say that a finite, nonempty subset X in Rn
carries a spherical t-design if there is a similarity
transformation mapping X to points on the unit sphere Sn1so that
1
|Y|
yY
f(y) =1
n
Sn1
f(y)d(y).
holds, for all polynomials f R[x1, . . . , xn] up to degree t. If the vertices of P form a 2-design, then they define an
eutactic form.
Definition: A Delaunay polytope P is called strongly perfect ifits vertex set form a 4-design.
Theorem: A strongly perfect Delaunay polytope P is perfect.
Proof: If f(v) = 0 for v a vertex of P then
1
n
S
n1
f(x)2dx =1
| vert P| vvertP
f(v)2 = 0
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II. Covering maxima
and characterization
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Lattice covering
We consider covering ofRn by n-dimensional balls of thesame radius, whose center belong to a lattice L.
The covering density has the expression
(L) =(L)nndet(L)
1
with (L) being the largest radius of Delaunay polytopes andn the volume of the unit ball B
n.
The only general method for computing (L) is to compute
all Delaunay polytopes ofL
.
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Covering maxima and pessima
Definition: If f Erdahl>0(n) then we define theinhomogeneous Hermite minimum by
H(f) = f(det Qf)1/n
We then have if Q is a quadratic form for L
(L)2 = max{f : f Erdahl(n) and Qf = Q}
(L) = n max{H(f)n/2 : f Erdahl(n) with Qf = Q}
Definition: A lattice L is said to be a covering maximum if forevery lattice L, which is near L we have (L) > (L).
Definition: A lattice L is said to be a covering pessimum if for
every lattice L, which is near L we have (L) > (L) exceptin a set of measure 0.
Theorem: We have equivalence between: L is a covering maximum (resp. pessimum). For every f Erdahl(n) with f = Q and (L)
2 = f the
function H(f) is a covering maximum (resp. pessimum)
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Convexity results
Theorem: The function f f is convex on Erdahl(n). Proof: Write f as
f(x) = Qf[x cf] f
and we havef = min
xRnf(x)
which implies the result.
Theorem (Minkovski): The function f
df
= (det Qf
)1/n
is convex on Erdahl(n).
But the product f H(f) = fdf is not convex.
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Characterization of covering maxima
Theorem: Take f Erdahl>0(n). The following areequivalent:
f is a covering maximum for H. f is perfect and eutactic.
Proof: Suppose that f is perfect and eutactic. We have thedifferentiation formula:
H(f+ f) = H(f) 1(det Qf)1/n
uf, f + o(f)
We also have
uf, f =
vZ(f)v(f)(v)
with v > 0. Iff is not collinear to f then there existsv Z(f) with (f)(v) > 0. Thus H decreases in direction f.
The reverse implication follows by preceding convexity results.
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Characterization of covering pessima
Theorem: Take f Erdahl>0(n). The following areequivalent:
f is a covering pessimum for H f is weakly eutactic
Proof: Suppose that f is weakly eutactic. We have thedifferentiation formula:
H(f+ f) = H(f) 1(det Qf)1/n
uf, f + o(f)
We also have
uf, f =
vZ(f)v(f)(v)
with v 0. Iff is not collinear to f and chosen outside ofa set of measure 0 then (f)(v) > 0 for all v and uf, f > 0.
The reverse implication follows by preceding convexity results.
T l l M f
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Topological Morse function property
Call Symm(n) the space of n n symmetric matrices.
Theorem: Take a lattice L and assume that all the Delaunaypolytopes of maximal circumradius are eutactic. The followingare equivalent: L is a non-degenerate topological Morse point for There exist a proper subspace SPL of Symm(n) such that for
every f0 Erdahl(n) with Qf0 = Q0 and (L)2
= f0 we haveQ Symm(n) : f Erdahl(n),
Qf = Q and Z(f) = Z(f0)
SPL
Theorem: The covering density function Q (Q) is a
topological Morse function if and only if n 3.
Proof: The lattice D4 has three translation classes ofDelaunay polytopes with the subspaces having zerointersection and their sum spanning Symm(4).
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III. Examples
K f D l l
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Known perfect Delaunay polytopes
In dimension 6, there is only Gossets 221, which defines theDelaunay polytopes of E6.
No classification in higher dimension. For n = 7, conjectured only two cases:
Lattice # vertices # facets | Sym |E7 56 702 2903040 (Gossets 321)
ER7 35 228 1440 For n = 8, a conjectured list of 27 possibilities. For n = 9, at least 100000 perfect Delaunay polytopes.
Some infinite series were built in M. Dutour, Infinite serie of extreme Delaunay polytopes,
European J. Combin. 26 (2005) 129132. V.P. Grishukhin, Infinite series of extreme Delaunay polytopes,
European J. Combin. 27 (2006) 481495. R. Erdahl and K. Rybnikov, An infinite series of perfect
quadratic forms and big Delaunay simplices in Zn, Proc.Steklov Inst. Math. 239 (2002) 159167.
E l f i i d i i
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Example of covering maxima and minima
The following lattices are covering maxima:name design strength # vertices # orbits Delaunay polytopes
E6 4 27 1
E7 5 56 2ER7 0 35 4O10 3 160 6
BW16 5 512 4O23 7 94208 523 7 47104 709
The following lattices are covering pessima:name design strength # vertices # orbits Delaunay polytopesZn 3 2n 1Dn 3 2n1 2 (or 1)E6 2 9 1E7 3 16 1E8 3 16 2
K12 3 81 4 Proof method: Take L a lattice
Enumerate all Delaunay polytopes of L up to symmetry Select the ones of maximum circumradius. Check for perfection (linear system problem)
check for eutacticity, resp. weakly eutacticity (linearprogramming problem)
I fi it f i i
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Infinite sequence of covering maxima
We need to find explicit lattice.
If n even, n 6, there is a n-dimensional extreme Delaunay
EDn formed with 3 layers of Dn1 lattice a vertex the n 1 half-cube the n 1 cross-polytope
If n odd, n 7, there is a n-dimensional extreme Delaunay
EDn formed with 3 layers of Dn1 lattice the n 1 cross polytope the n 1 half-cube the n 1 cross polytope
ED6 is Gossets 221 and ED7 is Gossets 321.
For a Delaunay polytope P, we define L(P) to be the smallestlattice containing P.
Theorem: The lattice L(EDn) spanned by EDn is a coveringmaxima.
We need the full list of orbits of Delaunay of L(EDn).
Proof method
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Proof method
Suppose that we have p orbits O1, . . . ,Op of Delaunaypolytopes P1, . . . ,Pp then
1
pi=1
|Oi| vol Pi.
If equality then the list O1, . . . ,Op is complete.
Take P is a polytope c P a point invariant under Aut P. IfF1, . . . , Fr are r orbits Orb1, . . . ,Orbr of facets of P then wehave
vol P r
i=1
|Orbi| vol(conv(Fi, c))
If equality then the list F1, . . . Fr is complete.
We get a conjectured list of orbits of Delaunay polytope ofL(EDn), of facets of EDn and prove their completude by the
volume formula.
Non generating perfect Delaunay polytopes
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Non generating perfect Delaunay polytopes
Theorem: There exist perfect Delaunay polytopes P in lattices
L with L(P) = L. Proof:
Take 2 s and 4s n and define the n-polytope
J(n, s) = {x {0, 1}n+1 such thatn+1
i=1
xi = s}
Define the vector vn,s =
14
4s, 0n+14s
.
P(n, s) = J(n, s) 2vn,s J(n, s) is a perfect n-dimensionalDelaunay polytope in Ln,s = L(P(n, s)).
Fact: If6s