BMS Course

Discrete Geometry II:Polytopal Geometry

http://www.math.tu-berlin.de/˜schmitt/L/10W/dg2/

— Lecture Notes, without guarantee —— I am happy about any feedback, corrections, suggestions for improvement, etc. —

— version of January 30, 2011 —

Prof. Gunter M. ZieglerInstitut fur Mathematik, MA 6-2

TU Berlin, 10623 BerlinTel. 030 314-25730

ziegler@math.tu-berlin.dehttp://www.math.tu-berlin.de/˜ziegler

TU Berlin, Winter term 2010/2011

Discrete Geometry II: Polytopal GeometryThis course is an introduction to Discrete Geometry. It will start with basic concepts and results in anarea, where combinatorial and convex-geometric methods meet in a great variety of ways. The focusof the exposition will be on an area that is very large, active, and quite relevant for different types ofapplications: Convex Polytopes. The introduction will include:– the study of interesting example (3-dimensional polytopes, cyclic polytopes, stacked polytopes, regu-

lar polytopes, associahedra, 4-dimensional polytopes, non-rational polytopes, etc.)– basic constructions (sum and product, projections, duality, projections, etc.)– development of structure theory (faces, face lattice, combinatorial type, realization space, etc.)– introduction to important methods (linear algebra, graph-theoretical methods, Schlegel diagrams, Gale

diagrams, etc.)– overview about the most important results in the area (about the computation of polytopes, diameter

bounds, realization and universality, polytopes with few vertices, face vectors and the g-Theorem, etc.)– discussion of some active problems and research questions (current hot topic: the Hirsch conjecture!).The course will include an introduction to and exercises in the polymake software system for theconstruction and analysis of polytopes.

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Discrete Geometry II: Polytopal Geometry — TU Berlin — BMS Course Winter 2010/2011 — Notes version: January 30, 2011 — Gunter M. Ziegler

Preface

This course is called “Discrete Geometry II” because it is in sequence with the “Discrete Geometry I:Computational Geometry” course taught by Carsten Schultz last term. However, the present course willhave a different point of view, and be largely independent.Here is a plan for the first lectures:

0. Some Basic Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Oct. 18

1. Combinatorial convexity (cf. Matousek [1, Chap. 1]) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Oct. 25

2. Polytopes and their representations (cf. Ziegler [2, Lect. 1]) . . . . . . . . . . . . . . . . . . . . . . . . . Nov. 1–8

3. Faces and the face lattice (cf. Ziegler [2, Lect. 2]) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Nov. 15

4. Graphs of polytopes; the Hirsch conjecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Nov. 22

5. Fundamental constructions (product, sum, wedges, stacking) . . . . . . . . . . . . . . . . . . . . . . . . . Nov. 29

6. Graphs of 3-polytopes; Steinitz’ theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Dec. 6

7. Rigidity theory, I (Bernd Schulze) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Dec. 13

Rigidity theory, II (Bernd Schulze) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Jan. 3

Rigidity theory, III (Bernd Schulze) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Jan. 10

Rigidity theory, IV (Bernd Schulze) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Jan. 17

8. f -vectors, shellability, and the Euler-Poincare equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Jan. 25

Lower bound and upper bound theorems, g-theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Jan. 31

9. Diameter and the Hirsch conjecture (Moritz Schmitt) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Feb. 7

[Oral exams] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Feb. 14

References

[1] Jirı Matousek. Lectures on Discrete Geometry, volume 212 of Graduate Texts in Math. Springer-Verlag, NewYork, 2002.

[2] Gunter M. Ziegler. Lectures on Polytopes, volume 152 of Graduate Texts in Mathematics. Springer-Verlag,New York, 1995. Revised edition, 1998; seventh updated printing 2007.

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Discrete Geometry II: Polytopal Geometry — TU Berlin — BMS Course Winter 2010/2011 — Notes version: January 30, 2011 — Gunter M. Ziegler

0 Some Basic Examples

Convex sets in Rd

Convex polytopes:V-polytope: Convex hull of a finite set of pointsH-polytope: Bounded intersection of finite set of closed halfspacesFaces:

• vertices

• edges

• ridges

• facets

Face numbersExamples:

1. The d-dimensional simplex

∆d = conv{0, e1, . . . , ed} = {x ∈ Rd : xi ≥ 0, x1 + · · ·+ xd ≤ 1}fk(∆d) =

(d+1k+1

)for −1 ≤ k ≤ d

2. The d-dimensional cube

fk(Cd) =(dk

)2d−k for 0 ≤ k < d

3. The d-dimensional cross polytope

fk(C4d ) =(

dk+1

)2k+1 for −1 ≤ k ≤ d

4. Cyclic polytopes Cd(n)

. . . an n-vertex d-dimensional polytope with the maximal number of facets (the “Upper boundtheorem”)

5. Stacked polytopes

. . . an n-vertex d-dimensional simplicial polytope with the minimal number of facets (the “Lowerbound theorem”)

6. The associahedron

. . . a d-dimensional polytope whose vertices correspond to the bracketings of a word with d + 2letters.

f0(Assd) = 1d+2

(2d+2d+1

)fd−1(Assd) =

(d+2

2

)− 1.

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1 Combinatorial convexity

1.1 Convex sets, Radon’s lemma and Caratheodory’s theorem

Equivalent definitions of a set C ⊆ Rd to be convex:• “Intuitively, a set is convex if its surface has no “dips”:” [2, p. 5]• x, y ∈ C implies x+ λ(y − x) ∈ C for λ ∈ [0, 1]• x1, . . . , xk ∈ C implies λ1x1 + · · ·+ λkxk ∈ C for k ≥ 1, λ1, . . . , λk ≥ 0, λ1 + · · ·+ λk = 1.

Exercise: A closed set is convex if and only if for x, y ∈ C it also contains 12(x+ y)

Equivalent definitions of the convex hull of a set X ⊆ Rd:• conv(X) := {λ1x1 + · · ·+λkxk : k ≥ 1, x1, . . . , xk ∈ X, λ1, . . . , λk ≥ 0, λ1 + · · ·+λk = 1}.• conv(X) is the smallest convex set that contains X .• conv(X) is the intersection of all convex sets that contain X .

A (finite) set X ⊂ Rd is in convex position if no x ∈ X is contained in the convex hull of X \ x.Any affinely independent set X ⊂ Rd is in convex position.An affinely independent set in Rd has cardinality at most d+ 1.

Lemma 1.1 (Radon’s lemma 19231). Any affinely dependent setA ⊆ Rd (in particular, every set of d+2points) can be partitioned into two disjoint subsets A1, A2 ⊂ A with convA1 ∩ convA2 6= ∅.

Proof. Analyze an affine dependence: The sum of coefficients is 0, so let A1 be the points with negativecoefficients, and A2 all the others.

Lemma 1.2 (Caratheodory’s theorem 19072). Any point in the convex hull of a set X ⊆ Rd is a convexcombination of at most d+ 1 points in X .

Proof. Analyze a minimal representation. If it needs more than d + 1 points, subtract multiples of anaffine dependence.

In the case of finite X , geometric interpretation from a triangulation of X , which may be constructedinductively.

1.2 Separation theorems

Theorem 1.3 (Separation theorem). If C,D ⊂ Rd are disjoint compact (bounded, closed) convex sets,then there is a hyperplane that separates them.

Proof. By compactness, there are points p ∈ C and q ∈ D such that the distance between C and D isrealized by them. Consider the bisector of the connecting line.

Important special case: when C is a polytope given by inequalities, and D is a point. For this case, wewill get the separation theorem as a consequence of the “Farkas lemma”.Be careful with proofs. For example, the proof given by Matousek [2, Thm. 1.2.4] is badly wrong (seethe errata webpage for this book).General (also: infinite-dimensional) versions are known under the name of “Hahn–Banach Theorem”(which was proved by Helly, 15 years before Hahn).

1Johann Radon: *16. Dezember 1887 in Decın, Bohemia; † 25. Mai 1956 in Wienhttp://en.wikipedia.org/wiki/Johann_Radon

2Constantin Caratheodory: *13. Sept. 1873 in Berlin; † 2. February 1950 in Munichhttp://en.wikipedia.org/wiki/Constantin_Caratheodory

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1.3 Helly’s theorem

Theorem 1.4 (Helly’s theorem 19133). Let C1, . . . , Cn ⊂ Rd be convex sets, n ≥ d + 1. Assume thatthe intersection of any d+ 1 of them is non-empty. Then the intersection of all of them C1 ∩ · · · ∩ Cn isnon-empty.

Proof. By induction on n, starting at n = d+ 1, using Radon’s lemma: Consider a minimal counterex-ample C1, . . . , Cn, where n ≥ d+ 2. Construct a1, . . . , ad+2, where ai lies in the intersection of all setsCj with j 6= i. Then consider the Radon point of a1, . . . , ad+2.

Note: Helly’s theorem is not true for infinitely many sets, if these are not necessarily compact.

1.4 Centerpoint theorem and Birch’s theorem

Definition 1.5. A centerpoint for a finite set X ⊂ Rd, |X| = n is a point c ∈ Rd such that any closedhalfspace that contains c also contains at least n

d+1 points of X .

A centerpoint is not usually unique, and it usually is not a point in X . (Example: if X is in convexposition!)

Theorem 1.6 (Centerpoint theorem: B. Neumann [3] 19454). R. Rado [4] 19465] Every finite set X ofn points in Rd has a centerpoint.

Proof. Equivalently, every open halfspace H that contains more than dd+1n points from X should con-

tain c.We would like to apply Helly’s theorem, but we can’t apply this to infinitely many open unboundedsubsets.Thus, for each such halfspaceH consider the (compact) subset CH := conv(H ∩X), and apply Helly’stheorem to these sets — there are even only finitely many different ones.

Remark: We have indeed established that there is a centerpoint c such that any convex set that containsmore than d

d+1n points from X does contain c.

Theorem 1.7 (Birch’s theorem 1959 [1]). Any 3N points in the plane can be partitioned into 3N triples,whose convex hulls intersect.

Birch’s proof. Find a centerpoint, order the points cyclically around this point, and partition them intotriples well-balanced from there.

Remark. This is now known as “Tverberg’s theorem”, since Birch conjectured a corresponding d-dimensional version, which however he could not prove — it was proved by Helge Tverberg in 1966[5]. Note that the centerpoint theorem is valid in the d-dimensional version, but it does not yield ad-dimensional version of Birch’s theorem, since there is no d-dimensional analogue for “cyclic orderaround a point”.Birch gives a topological proof for the centerpoint theorem: “so far as I know, it is new. The plane caseof the lemma was proved in 945 by Neumann (5); his proof, though elementary, is long, and does notextend to higher dimensions.”

3Eduard Helly: *1. June 1884 in Wien; † 28. November 1943 in Chicagohttp://www-history.mcs.st-andrews.ac.uk/Biographies/Helly.html

4Bernhard H. Neumann *15 October 1909 in Berlin; † 21 October 2002 in Canberra, Australiahttp://en.wikipedia.org/wiki/Johann_Radon

5Richard Rado *28 April 1906 in Berlin; † , Died: 23 December 1989 in Henley-on-Thames, Oxfordshire, Englandhttp://www-history.mcs.st-andrews.ac.uk/Biographies/Rado_Richard.html

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References

[1] B. J. Birch. On 3N points in a plane. Math. Proc. Cambridge Phil. Soc., 55:289–293, 1959.

[2] Jirı Matousek. Lectures on Discrete Geometry, volume 212 of Graduate Texts in Math. Springer-Verlag, NewYork, 2002.

[3] Bernhard H. Neumann. On an invariant of plane regions and mass distributions. J. London Math. Soc.,20:226–237, 1945.

[4] Richard Rado. A theorem on general measure. J. London Math. Soc., 21:291–300, 1946.

[5] Helge Tverberg. A generalization of Radon’s theorem. J. London Math. Soc., 41:123–128, 1966.

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2 Polytopes and their representations

(cf. Ziegler [1, Lect. 1])

2.1 The “Main Theorem”

Definition 2.1 (H-polyhedron). An H-polyhedron is a set P = P (A, z) = {x ∈ R : Ax ≤ z} forA ∈ Rm×d: the solution set of a finite system of linear inequalities.

Definition 2.2 (V-polyhedron). A V-polyhedron is a set P = conv(V ) + cone(Y ) for V ∈ Rd×k,Y ∈ Rd×n: a vector sum (“Minkowski sum”) of a finite convex hull and a finitely-generated convex conefinitely-generated (“polyhedral”) cone cone(Y ) = {Y t : t ≥ 0} for Y ∈ Rd×n.

Lemma 2.3. A V-polytope is a V-polyhedron that does not contain an unbounded ray.

Definition 2.4. AnH-polytope is anH-polyhedron that does not contain an unbounded ray.

Theorem 2.5 (Main Theorem for Polytopes). A subset P ⊆ Rd is a V-polytope if and only if it is anH-polytope.

Corollary 2.6.(i) Any intersection of a polytope with an affine subspace is a polytope.

(ii) Any intersection of a polytope with a polytope is a polytope.(iii) Any Minkowski sum of polytopes is a polytope.(iv) Any projection image of a polytope is a polytope.

Theorem 2.5 follows from the Main Theorem for Polyhedra:

Theorem 2.7 (Main Theorem for Polyhedra). A subset P ⊆ Rd is a V-polyhedron if and only if it is anH-polyhedron.

Corollary 2.8. AnH-polyhedron is bounded if and only if it does not contain an unbounded ray.

Theorem 2.9 (Main Theorem for Cones). A subset C ⊆ Rd is a V-cone if and only if it is anH-cone.

Proof that the Main Theorem for Cones implies the one for polyhedra. For anyH-polyhedronP = P (A, z),we get anH-cone

C(P ) := P((−1 O−z A

),

(00

)).

in the halfspace where the first coordinate is nonnegative.For any V-polyhedron P = conv(V ) + cone(Y ) we get a V-cone

C(P ) := cone(

1l OV Y

).

whose generators have the first coordinate nonnegative.In both cases we see that

P = {x ∈ Rd :

(1x

)∈ C(P )}.

whereC(P ) is anH-cone contained in {x ∈ Rd+1 : x0 ≥ 0} if and only if P is anH-polyhedron, andC(P ) is a V-cone contained in {x ∈ Rd+1 : x0 ≥ 0} if and only if P is a V-polyhedron.

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2.2 Fourier–Motzkin elimination

Proof of Theorems 2.7/2.9. Grand plan:(1) Every V-polyhedron can be written as a projection image of anH-polyhedron (simplex + orthant).(2) Fourier–Motzkin elimination shows that projection images ofH-polyhedra areH-polyhedra.

(3) EveryH-polyhedron can be written as an affine section of a V-polyhedron (a cone).(4) Fourier–Motzkin elimination shows that affine sections of V-polyhedra are V-polyhedra.

Some Details:(1) Every V-polyhedron

conv(V ) + cone(Y ) ={x ∈ Rd : ∃ t ∈ Rn,u ∈ Rn′

: x = V t + Y u, t ≥ 0, u ≥ 0, 1l t = 1}

can be interpreted as the projection of a set

{

xtu

∈ Rd+n+n′: x = V t + Y u, t ≥ 0, u ≥ 0, 1l t = 1

}that is anH-polyhedron.(2) . . . illustrated on an example. Note this is conceptual, but also provides an algorithm.(3) EveryH-polyhedron

P (A, z) = {x ∈ Rd : Ax ≤ z}

can be written as the intersection of a cone

C0(A) := {(x

w

)∈ Rd+m : Ax ≤ w}

with an affine coordinate subspace

{(x

w

)∈ Rd+m : w = z}.

The cone C0(A) can be written as

C0(A) := {(

x

Ax

): x ∈ Rd}+ {

(0

w′

): w′ ∈ Rm,w′ ≥ 0},

which is a linear subspace plus an orthant, both of which are easy to write as V-polyhedra.(4) . . . sketch for affine situation.

2.3 Fourier–Motzkin elimination for cones

Algebraic proof of Theorem 2.9.Every V-cone is an H-cone: Here the key point is that if P = P (A,0), then the projected cone (“lastcoordinate deleted”) can be described as P /d = P (CA,0), where C is a non-negative matrix (with atmost two non-zero entries per row).Every H-cone is a V-cone: Here assume that P = cone(Y ) contains a point v with vd = 0. Now eitherwe have tiydi = 0 for all i, in which case we get v ∈ cone({yi : ydi = 0}), or we can expand vd = 0, toget

Λ :=∑

i:ydi>0

tiydi =∑

j:ydj<0

tj(−ydj) > 0.

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With this, we can rewrite v as

v =∑

i:ydi=0

tiyi +∑

i:ydi>0

tiyi +∑

j:ydj<0

tjyj

=∑

i:ydi=0

tiyi +1

Λ

∑i:ydi>0

( ∑j:ydj<0

tj(−ydj)

)tiyi +

1

Λ

∑j:ydj<0

( ∑i:ydi>0

tiydi

)tjyj

=∑

i:ydi=0

tiyi +∑

i:ydi>0j:ydj<0

titjΛ

((−ydj)yi + ydiyj

).

For more details see [1, Sect. 1.3].

2.4 The Farkas Lemmas

There are many variants of the “Farkas Lemma”, as• theorems of the alternative• transposition theorems• duality theorems• good characterizations• certificates of validity, or• separation theorems.

Proposition 2.10 (Farkas Lemma I). Let A ∈ Rm×d and z ∈ Rm.Either there exists a point x ∈ Rd with Ax ≤ z,or there exists a row vector c ∈ (Rm)∗ with c ≥ O, cA = O and cz < 0,but not both.

Proof. First observe that both conditions cannot hold at the same time: otherwise there are a columnvector x ∈ Rd and a row vector c ∈ (Rm)∗ with

0 = Ox = (cA)x = c(Ax) ≤ cz < 0,

which is a contradiction. Similar calculations are valid for all other versions of the Farkas lemma.Now assume that P := P (A, z) is empty, i.e., Q := P

((−z, A),0

)does not contain a point with

with x0 > 0. Here Q is an H-cone. Now we eliminate the variables x1, . . . , xd from Q, to get to a 1-dimensional situation. If the resulting cone is contained in {x0 ≤ 0} then the eliminated system, whichis of the form P

(C(−z, A),0

)with a non-negative matrix C, must contain one single inequality that

guarantees that. Pick the corresponding row of C.

Proposition 2.11 (Farkas Lemma II). Let A ∈ Rm×d and z ∈ Rm.Either there exists a point x ∈ Rd with Ax = z, x ≥ 0,or there exists a row vector c ∈ (Rm)∗ with cA ≥ O and cz < 0,but not both.

Proof. Rewrite the system of linear equations in non-negative variables as a system of inequalities – andapply Farkas Lemma I.

Proposition 2.12 (Farkas Lemma III). Let A ∈ Rm×d, z ∈ Rm, a0 ∈ (Rd)∗, and z0 ∈ R.Then a0x ≤ z0 is valid for all x ∈ Rd with Ax ≤ z, if and only if(i) there exists a row vector c ≥ O such that cA = a0 and cz ≤ z0, or

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(ii) there exists a row vector c ≥ O such that cA = O and cz < 0,or both.

Proposition 2.13 (Farkas Lemma IV). Let V ∈ Rd×n, Y ∈ Rd×n′, and x ∈ Rd.

Either there exist t,u ≥ 0 with 1l t = 1 and x = V t + Y u,or there exists a row vector (α,a) ∈ (Rd+1)∗ with avi ≤ α for all i ≤ n, ayj ≤ 0 for all j ≤ n′, whileax > α,but not both.

2.5 Lineality Space, Recession Cone, Homogenization

Using the Farkas lemma, we can give an invariant description of some very important constructions(notably the recession cone and the homogenization of a convex set) and establish their basic properties.In Proposition 2.17 we will see that the homogenization homog(P ) of a polyhedron coincides with the“associated cone” C(P ) that we used in Section 2.1.

Definition 2.14. Let P ⊆ Rd be a convex set. Then the lineality space of P is defined as

lineal(P ) := {y ∈ Rd : x + ty ∈ P for all x ∈ P, t ∈ R},

and the recession cone of P is defined as

rec(P ) := {y ∈ Rd : x + ty ∈ P for all x ∈ P, t ≥ 0}.

Proposition 2.15. Let P ⊆ Rd be a convex set.

(i) If P = P (A, z) is anH-polyhedron, then so is its recession cone:

rec(P ) = P (A,0).

(ii) If P = conv(V ) + cone(Y ) is a V-polyhedron, then so is its recession cone:

rec(P ) = cone(Y ).

Definition 2.16. Let P ⊆ Rd be a convex set. Then the homogenization of P is defined as

homog(P ) := {t(

1x

): x ∈ P, t > 0} + {

(0y

): y ∈ rec(P )}.

Proposition 2.17. Let P ⊆ Rd be a convex set.

(i) If P = P (A, z) is anH-polyhedron, then its homogenization is also anH-polyhedron:

homog(P ) = P((−1 O−z A

),

(00

))= C(P ).

(ii) If P = conv(V ) + cone(Y ) is a V-polyhedron, then so is its homogenization:

homog(P ) = cone(

1l OV Y

)= C(P ).

– see also [1, Sect. 1.5]

References

[1] Gunter M. Ziegler. Lectures on Polytopes, volume 152 of Graduate Texts in Mathematics. Springer-Verlag,New York, 1995. Revised edition, 1998; seventh updated printing 2007.

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3 Faces and the face lattice

(cf. Ziegler [1, Lect. 2])

3.1 Vertices, faces, and facets

Definition 3.1. Let P ⊆ Rd be a convex polytope. A linear inequality cx ≤ c0 is valid for P if it issatisfied for all points x ∈ P . A face of P is any set of the form

F = P ∩ {x ∈ Rd : cx = c0}

where cx ≤ c0 is a valid inequality for P . The dimension of a face is the dimension of its affine hull:dim(F ) := dim(aff(F )).

P is a face of P , all other faces are called proper.∅ is also a face of P , all other faces are called non-trivial.The faces of dimensions 0, 1, dim(P )− 2, and dim(P )− 1 are called vertices, edges, ridges, and facets,respectively. The set of all vertices of P , the vertex set, is denoted by vert(P ).

Proposition 3.2. Every polytope P is the convex hull of its vertices, P = conv(vert(P ))

Proposition 3.3. Let P be a polytope, F a face of P .(i) F is a polytope with vertex set vert(F ) = F ∩ vert(P )

(ii) Every intersection of faces is a face.(iii) The faces of F are exactly the faces of P contained in F .(iv) F = P ∩ aff(F )(v) If F ⊂ G are distinct faces, then dim(F ) < dim(G)

(vi) P has only finitely many faces.

Definition 3.4. If P is a polytope and v is a vertex of P , then the vertex figure of P at v is the intersectionof P with a hyperplane that has v on one side and all other vertices of P on the other side.

Problem. Show that the vertex figure is well-defined up to projective equivalence.Describe construction rules for a vertex figure that is more unique.

Proposition 3.5. The (k − 1)-dimensional faces of a vertex figure P/v = P ∩H are in bijection withthe k-faces of P .

3.2 The face lattice

Poset (S,≤): finite partially ordered set (relation reflexive, transitive, antisymmetric)chain, lengthintervalboolean posetbounded poset: minimal element 0, maximal element 1graded poset: bounded, and all maximal chains have the same ranklattice: bounded, minimal upper bounds (join, ∨) and maximal lower bounds (meet, ∧) existatoms, coatoms, atomic and coatomic latticeorder dual Sop

Hasse diagram, draws cover relations.

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Definition 3.6. The face lattice L(P ) of a polytope P is the poset of faces, ordered by inclusion.Two polytopes are combinatorially equivalent, P ' Q if their face lattices are isomorphic, P ∼= Q.

Theorem 3.7. Let P be a convex polytope.(i) L(P ) is a graded lattice, r(F ) = 1 + dim(F )

(ii) Every interval is the face lattice of a polytope.(iii) The “diamond property”(iv) The opposite L(P )op is also a polytope face lattice.(v) L(P ) is atomic and coatomic.

Here part (iv) is deferred to the next section, it is proved using the polar of a polytope.

3.3 Polarity

The relative interior relint(P ) of a polytope P is the set of all points in P that are not contained in aproper subset.Thus we have a decomposition into disjoint subsets

P =⊎

F∈L(P )

relint(F )

Definition 3.8. For any subset P ⊆ Rd, the polar set is a subset of the dual space (!) given by

P∆ := {c ∈ (Rd)∗ : cx ≤ 1 for all x ∈ P} ⊆ (Rd)∗.

The following theorem characterizes polar sets. Note that we get the result we want, “the polar polytopehas the opposite face lattice” only in the special case where the origin is in the relative interior. Indeed,if we want that P∆ is a polytope, then we need that the origin is in the interior, and indeed the polytopeis full-dimensional.

Theorem 3.9. (i) P ⊆ Q implies P∆ ⊇ Q∆ and P∆∆ ⊆ Q∆∆,(ii) P ⊆ P∆∆,

(iii) P∆ and P∆∆ are convex,(iv) O ∈ P∆, and 0 ∈ P∆∆,(v) if P is a polytope and 0 ∈ P , then P = P∆∆,

(vi) if a polytope P with 0 ∈ int(P ) is given by P = conv(V ), then

P∆ = {a : aV ≤ 1l },

(vii) if a polytope P with 0 ∈ int(P ) is given by P = P (A,1), then

P∆ = {cA : c ≥ O, c1 = 1}.

Corollary 3.10. If P ⊂ Rd has the origin in its interior, then L(P∆) ∼= L(P )op.

Indeed, we obtain precise characterizations of the face F♦ ⊂ P∆ that corresponds to F ⊆ P with respectto this correspondence:

Proposition 3.11. Assume that P = conv(V ) = P (A,1) is a polytope in Rd, and that

F = conv(V ′) = {x ∈ Rd : A′′x ≤ 1, A′x = 1}

is a face of P , with V = (V ′, V ′′) and A =(A′

A′′

), where the columns of V ′ are the points on F , and the

inequalities given by the rows of A′ correspond to the inequalities that are tight on F . Then

F♦ = {c′A′ : c′ ≥ O, c′1 = 1} = {a : aV ′′ ≤ 1l , aV ′ = 1l }.

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3.4 Simple and simplicial polytopes

Definition of simple and of simplicial polytopes.Many equivalent characterizations.Examples

3.5 Projective transformations

See [1, Sect. 2.6].

References

[1] Gunter M. Ziegler. Lectures on Polytopes, volume 152 of Graduate Texts in Mathematics. Springer-Verlag,New York, 1995. Revised edition, 1998; seventh updated printing 2007.

13

4 Graphs of polytopes

(cf. Ziegler [6, Lect. 3])

4.1 The graph of a polytope

Definition 4.1. The graph G(P ) of a polytope P is the abstract graph whose vertices are given by the0-faces and whose edges are given by the 1-faces of the polytope.

– abstract (finite, simple) graph (artificial terminology: nodes, arcs)Problem 1: Characterize graphs of polytopes.For example, graphs of polytopes of dimension 3 will be characterized by Steinitz’ theorem (next chapter)Problem 2: Which polytope properties can be deduced from the graph?– cannot tell the dimension, since ∆5 and C4(6) ∼= ∆2 ⊕∆2 have the same graph K6.– cannot tell from the graph whether a d-polytope is simplicial.(Example for d = 4: pyramid over bipyramid over triangle)– a d-dimensional polytope is simple if and only if its graph is d-regular.– Is Peterson graph × Peterson graph of a polytope? (See recent paper by Pfeifle, Pilaud & Santos [5]!)Problem 3: Can reconstruct the combinatorics of the polytope from the graph?– even for simplicial 4-polytopes, the combinatorics cannot be derived from the graph. (No examplegiven in class.)Problem 4: Describe properties of polytope graphs.

Proposition 4.2. Let G = G(P ) be the graph of a d-polytope.(i) The graph G is d-connected: between any two vertices u, v, there are d paths that are disjoint

except for the ends. (“Balinski’s Theorem”)(ii) The graph G contains a subdivision of Kd+1 as a subgraph.

Proof. (i) proved in Theorem 4.6 below in stronger version.(ii) induction on d, via the vertex figure at an arbitrary vertex v, and Lemma 3.5.

Corollary 4.3. The graphs of d-polytopes are not planar for d ≥ 4.However, the graphs of d-polytopes are planar for d ≤ 3.

4.2 The directed graph of a linear program

Linear program: max /cx : x ∈ P for anH-polytope P .Without loss of generality/after a projective transformation, we may assume that the objective functionis cx = xd, and this is in general position (no horizontal edges).

Lemma 4.4. The neighbors span the cone at the vertex.

Proof. The edges at a vertex correspond to the vertices of the vertex figure. Simple calculation. Compare[6, Lemma 3.6].

Proposition 4.5. Every linear function in general position defines an acyclic orientation of the graphthat has a unique sink on each non-empty face.

14

Theorem 4.6 (Holt & Klee [3] directed version of Balinski’s theorem). If P is a d-polytope and cx is alinear function in general position, then there are d vertex-disjoint monotone edge-paths from the uniqueminimal vertex vmin to the unique maximal vertex vmax.

Proof. After a suitable projective (!) coordinate transformation, we may assume that cx = xd, vmin = 0,vmax = ed. We use Menger’s theorem, so let S be any set of at most d − 1 vertices of P differentfrom vmin, vmax. This set is contained in a vertical hyperplane G. After a further (affine) coordinatetransformation we may assume that the hyperplane G is given by xd−1 = 0 or by xd−1 = 1.Now analyze the situation in the 2-dimensional projection to the (xd−1, xd)-plane. Any edge path in thisprojection can be lifted to Rd, since any edge in the projection corresponds to some face in Rd, whichhas unique minimal and maximal vertex and a monotone edge path from one to the other.For details, see [3, Prop. 2.1].

4.3 How to tell a simple polytope from its graph

Kalai’s proof for the

Theorem 4.7 (Blind–Mani [1]; Kalai [4]; Friedman [2]). Given the graph of a simple polytopes, thesubgraphs corresponding to facets can be reconstructed — easily (simple proof: Kalai), and fast (“inpolynomial time”: Friedman).

Proof. See [6, Sect. 3.4].

References

[1] Roswitha Blind and Peter Mani-Levitska. On puzzles and polytope isomorphisms. Aequationes Math.,34:287–297, 1987.

[2] Eric J. Friedman. Finding a simple polytope from its graph in polynomial time. Discrete Comput. Geometry,41:249–256, 2009.

[3] Fred Holt and Victor Klee. A proof of the strict monotone 4-step conjecture. In B. Chazelle, J. E. Goodman,and R. Pollack, editors, Advances in Discrete and Computational Geometry (South Hadley, MA, 1996), volume223 of Contemporary Mathematics, pages 201–216, Providence RI, 1998. Amer. Math. Soc.

[4] Gil Kalai. A simple way to tell a simple polytope from its graph. J. Combinatorial Theory, Ser. A, 49:381–383,1988.

[5] Julian Pfeifle, Vincent Pilaud, and Francisco Santos. Polytopality and cartesian products of graphs. Preprint,September 2010, 21 pages, http://arxiv.org/abs/1009.1499.

[6] Gunter M. Ziegler. Lectures on Polytopes, volume 152 of Graduate Texts in Mathematics. Springer-Verlag,New York, 1995. Revised edition, 1998; seventh updated printing 2007.

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5 Examples

5.1 Cyclic polytopes

cyclic polytopes:Construction: moment curve, Caratheodory curvequestion: is there an order 3 curve in S2?Gale’s evenness condition

5.2 Fundamental constructions

1. pyramid:face lattice, f -vector, graph, dual

2. prism:dual to bipyramid

3. bipyramid:face lattice, f -vector, graph, dual, simplicial

4. product:vertex description, facet descriptionface lattice

5. (direct sum:dual to productExample: ∆2 ⊕∆2 is neighborly!

6. subdirect sum (a la McMullen):(P, F )⊕Q

7. vertex splitting:Example: odd-dimensional cyclic polytopes

8. wedges:dual to vertex splittingone more vertex, one more facetdiameter does not decrease

9. stacking:one more vertex, d− 1 more facetsExamples: stacked polytopes.

10. vertex truncation

5.3 An example: The hypersimplex

Definition 5.1 (hypersimplex). For 1 ≤ k ≤ d, the hypersimplex ∆d(k) in Rd+1 is

∆d(k) = conv{v ∈ {0, 1}d+1 :d+1∑i=1

vi = k}

= {x ∈ Rd+1 : 0 ≤ xi ≤ 1 for 1 ≤ i ≤ d+ 1,

d∑i=1

xi = k} ⊂ Rd+1.

16

It seems that the hypersimplices first appeared in Gabrielov, Gel’fand & Losik [1, Sect. 1.6] — in thetheory of characteristic classes. See also Gel’fand, Goresky, MacPherson & Serganova [2].Analysis:

1. full-dimensional version in Rd

∆′d(k) ∼= conv{v ∈ {0, 1}d : k − 1 ≤d∑

i=1

vi ≤ k}

= {x ∈ Rd : 0 ≤ xi ≤ 1 for 1 ≤ i ≤ d, k − 1 ≤d∑

i=1

xi ≤ k}

2. ∆d(k) ∼= ∆d(d+ 1− k)

3. standard simplex as ∆d = ∆d(1) ∼= ∆d(d).

4. octahedron ∆3(2) centrally symmetric for ∆2k−1(k)

5. ∆d(k) has(d+1k

)vertices, and 2d+ 1 facets, if 2 ≤ k ≤ d− 1

6. symmetry group transitive on vertices; 2 orbits of facets in general

7. facets of ∆d(k): d+ 1 facets of type ∆d−1(k), d+ 1 facets of type ∆d−1(k − 1).

8. ∆d(k) is d-simplicial (every 2-face is a triangle), and (d− 2)-simple (every edge in exactly d− 1facets — that is, the dual is 2-simplicial)

References

[1] Andrei M. Gabrielov, Izrail M. Gel’fand, and Mark V. Losik. Combinatorial computation of characteristicclasses. Functional Analysis Appl., 9:103–115, 1975.

[2] Izrail M. Gel’fand, Mark Goresky, Robert D. MacPherson, and Vera Serganova. Combinatorial geometries,convex polyhedra and Schubert cells. Advances in Math., 63:301–316, 1987.

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6 Graphs of 3-polytopes; Steinitz’ theorem

6.1 Steinitz’ theorem

Theorem 6.1 (Steinitz [3, 4]). The graphs of 3-polytopes are exactly the 3-connected planar graphs.

6.2 Proof strategies

Strategy 1 (Steinitz): local modifications, such as ∆Y -operations.. . . see [5, Lect. 4]Strategy 2 (Maxwell–Tutte): rubber bands: lifting a planar equilibrium representation.. . . see Richter-Gebert [2, Part. IV]Strategy 3 (Koebe–Thurston–Bobenko–Springborn): constructing circle patterns.. . . see Bobenko & Springborn [1] and [6, Lect. 1],ftp://ftp.math.tu-berlin.de/pub/combi/ziegler/WWW/99utah.pdf

References

[1] Alexander I. Bobenko and Boris A. Springborn. Variational principles for circle patterns, and Koebe’s theorem.Transactions Amer. Math. Soc., 356:659–689, 2004.

[2] Jurgen Richter-Gebert. Realization Spaces of Polytopes, volume 1643 of Lecture Notes in Mathematics.Springer-Verlag, Berlin Heidelberg, 1996.

[3] Ernst Steinitz. Polyeder und Raumeinteilungen. In W. Fr. Meyer and H. Mohrmann, editors, Encyklopadie dermathematischen Wissenschaften, Dritter Band: Geometrie, III.1.2., Heft 9, Kapitel III A B 12, pages 1–139.B. G. Teubner, Leipzig, 1922.

[4] Ernst Steinitz and Hans Rademacher. Vorlesungen uber die Theorie der Polyeder. Springer-Verlag, Berlin,1934. Reprint, Springer-Verlag 1976.

[5] Gunter M. Ziegler. Lectures on Polytopes, volume 152 of Graduate Texts in Mathematics. Springer-Verlag,New York, 1995. Revised edition, 1998; seventh updated printing 2007.

[6] Gunter M. Ziegler. Convex polytopes: Extremal constructions and f -vector shapes. In E. Miller, V. Reiner,and B. Sturmfels, editors, “Geometric Combinatorics”, Proc. Park City Mathematical Institute (PCMI) 2004,pages 617–691, Providence, RI, 2007. American Math. Society. With an appendix by Th. Schroder and N.Witte.

7 Rigidity Theory

. . . see separate Lecture Notes, by Bernd Schulze

18

8 f -vectors, lower bound and upper bound theorems, g-theorem

8.1 Euler-Poincare equation and shellability

Definition: f -vectorsTheorem: The Euler-Poincare equation for d-polytopesShellabilityProof of the Euler-Poincare equation, using shellability. See [6, Sect. 8.2].

8.2 f -vectors of polytopes for d = 3 and for d = 4

d = 3: complete answer, due to Steinitz [5]:

F3 = {(f0, f1, f2) ∈ Z3 : f1 = f0 + f2 − 2, f2 ≤ 2f0 − 4, f0 ≤ 2f2 − 4}.

d = 4: no complete answer available.The “upper bound theorem” for d = 4: f3 ≤ 1

2f0(f0 − 3), with equality if and only if the graph iscomplete (example: cyclic 4-polytopes).

Theorem 8.1 (see Grunbaum [3]).

F|(0,3) = {(f0, f3) ∈ Z2 : f3 ≤ 12f0(f0 − 3), f0 ≤ 1

2f3(f3 − 3)}.

The f -vector cone in coordinates

ϕ1 =f1 − 10

f0 + f3 − 10, ϕ2 =

f2 − 10

f0 + f3 − 10

Conditions:ϕ1, ϕ2 ≥ 1, |ϕ2 − ϕ1| ≤ 1, ϕ2 + ϕ1 ≥ 5

2 .

Problem: Is ϕ2 + ϕ1 (“fatness”) bounded for 4-polytopes?

8.3 Upper bound theorem and lower bound theorem

h-vector and g-theoremUpper bound theorem — McMullen [4]Proof: using shellability, see [6, Sect. 8.4]Lower bound theorem — Barnette [1, 2]Proof: using rigidity, see Chapter 7

8.4 The g-theorem

. . . in Bjorner’s version, using the M -matrices; see [6, Sect. 8.6]

References

[1] David W. Barnette. The minimum number of vertices of a simple polytope. Israel J. Math., 10:121–125, 1971.

[2] David W. Barnette. A proof of the lower bound conjecture for convex polytopes. Pacific J. Math., 46:349–354,1973.

19

[3] Branko Grunbaum. Convex Polytopes, volume 221 of Graduate Texts in Math. Springer-Verlag, New York,2003. Second edition prepared by V. Kaibel, V. Klee and G. M. Ziegler (original edition: Interscience, London1967).

[4] Peter McMullen. The numbers of faces of simplicial polytopes. Israel J. Math., 9:559–570, 1971.

[5] Ernst Steinitz. Uber die Eulerschen Polyederrelationen. Archiv fur Mathematik und Physik, 11:86–88, 1906.

[6] Gunter M. Ziegler. Lectures on Polytopes, volume 152 of Graduate Texts in Mathematics. Springer-Verlag,New York, 1995. Revised edition, 1998; seventh updated printing 2007.

20

9 Diameter and the Hirsch conjecture (Moritz Schmitt)

....

21

References

[1] A. BARVINOK, A Course in Convexity, vol. 54 of Graduate Studies in Math., Amer. Math. Soc.,Providence, RI, 2002.

[2] A. BRØNDSTED, An Introduction to Convex Polytopes, vol. 90 of Graduate Texts in Mathematics,Springer-Verlag, New York Berlin, 1983.

[3] J. E. GRAVER, Counting on Frameworks. Mathematics to Aid the Design of Rigid Structures, vol. 25of Dolciani Mathematical Expositions, Mathematical Assoc. America, Washington, DC, 2001.

[4] P. M. GRUBER, Convex and Discrete Geometry, vol. 336 of Grundlehren Series, Springer, 2007.

[5] B. GRUNBAUM, Convex Polytopes, vol. 221 of Graduate Texts in Math., Springer-Verlag, NewYork, 2003. Second edition prepared by V. Kaibel, V. Klee and G. M. Ziegler (original edition:Interscience, London 1967).

Software

Examples — computations can be done using the polymake software system by Michael Joswig andEwgenij Gawrilow, see http://www.polymake.de

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