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The Hirsch Conjecture Counter-examples to Hirsch Simplicial complexes Normal complexes

Combinatorial diameter of polytopes and abstractions of them

Francisco Santos

Departamento de Matemáticas, Estadística y Computación Universidad de Cantabria, Spain http://personales.unican.es/santosf

GdR Informatique Mathémaique, Paris — January 19, 2016 1

The Hirsch Conjecture Counter-examples to Hirsch Simplicial complexes Normal complexes The Hirsch Conjecture

The Hirsch conjecture

Conjecture (W. M. Hirsch, 1957) For every polyhedron P with n facets and dimension d, diam(P) ≤ n − d.

2

The Hirsch Conjecture Counter-examples to Hirsch Simplicial complexes Normal complexes The Hirsch Conjecture

The Hirsch conjecture

Conjecture (W. M. Hirsch, 1957) For every polyhedron P with n facets and dimension d, diam(P) ≤ n − d. Examples polytope facets dimension n − d diameter k-gon k 2 k − 2 ⌊k/2⌋ cube 6 3 3 3 dodecahedron 12 3 9 5

• ctahedron

8 3 5 2 n-cube 2n n n n

2

The Hirsch Conjecture Counter-examples to Hirsch Simplicial complexes Normal complexes The Hirsch Conjecture

Brief history of the conjecture

3

The Hirsch Conjecture Counter-examples to Hirsch Simplicial complexes Normal complexes The Hirsch Conjecture

Brief history of the conjecture

It was communicated by W. M. Hirsch to G. Dantzig in

• 1957. Hirsch knew it for n − d ≤ 3.

3

The Hirsch Conjecture Counter-examples to Hirsch Simplicial complexes Normal complexes The Hirsch Conjecture

Brief history of the conjecture

It was communicated by W. M. Hirsch to G. Dantzig in

• 1957. Hirsch knew it for n − d ≤ 3.

In 1967, Klee and Walkup found an unbounded counter-example (∃ P with d = 4, n = 8, diam(P) = 5).

3

The Hirsch Conjecture Counter-examples to Hirsch Simplicial complexes Normal complexes The Hirsch Conjecture

Brief history of the conjecture

It was communicated by W. M. Hirsch to G. Dantzig in

• 1957. Hirsch knew it for n − d ≤ 3.

In 1967, Klee and Walkup found an unbounded counter-example (∃ P with d = 4, n = 8, diam(P) = 5). Several special cases have been proved: d ≤ 3, 0/1-polytopes, n − d ≤ 6, . . .

3

The Hirsch Conjecture Counter-examples to Hirsch Simplicial complexes Normal complexes The Hirsch Conjecture

Brief history of the conjecture

It was communicated by W. M. Hirsch to G. Dantzig in

• 1957. Hirsch knew it for n − d ≤ 3.

In 1967, Klee and Walkup found an unbounded counter-example (∃ P with d = 4, n = 8, diam(P) = 5). Several special cases have been proved: d ≤ 3, 0/1-polytopes, n − d ≤ 6, . . . [S. 2012] contains the first bounded counter-examples (∃ P with d = 43, n = 86, diam(P) > 43).

3

The Hirsch Conjecture Counter-examples to Hirsch Simplicial complexes Normal complexes The Hirsch Conjecture

Brief history of the conjecture

It was communicated by W. M. Hirsch to G. Dantzig in

• 1957. Hirsch knew it for n − d ≤ 3.

In 1967, Klee and Walkup found an unbounded counter-example (∃ P with d = 4, n = 8, diam(P) = 5). Several special cases have been proved: d ≤ 3, 0/1-polytopes, n − d ≤ 6, . . . [S. 2012] contains the first bounded counter-examples (∃ P with d = 43, n = 86, diam(P) > 43). But: All known counterexamples have diameter only a small (∼1.05, ∼1.25) constant times the Hirsch bound.

3

The Hirsch Conjecture Counter-examples to Hirsch Simplicial complexes Normal complexes The Hirsch Conjecture

Brief history of the conjecture

It was communicated by W. M. Hirsch to G. Dantzig in

• 1957. Hirsch knew it for n − d ≤ 3.

In 1967, Klee and Walkup found an unbounded counter-example (∃ P with d = 4, n = 8, diam(P) = 5). Several special cases have been proved: d ≤ 3, 0/1-polytopes, n − d ≤ 6, . . . [S. 2012] contains the first bounded counter-examples (∃ P with d = 43, n = 86, diam(P) > 43). But: All known counterexamples have diameter only a small (∼1.05, ∼1.25) constant times the Hirsch bound. In the general case we do not even know of a polynomial bound for diam(P) in terms of n and d.

3

The Hirsch Conjecture Counter-examples to Hirsch Simplicial complexes Normal complexes The Hirsch Conjecture

Motivation: LP

The feasibility region P = {x ∈ Rd : Ax ≤ b}, A ∈ Rn×d

• f a linear program is a polyhedron P with (at most) n

facets and d dimensions.

4

The Hirsch Conjecture Counter-examples to Hirsch Simplicial complexes Normal complexes The Hirsch Conjecture

Motivation: LP

The feasibility region P = {x ∈ Rd : Ax ≤ b}, A ∈ Rn×d

• f a linear program is a polyhedron P with (at most) n

facets and d dimensions. Optimal value (if bounded) is always attained at a vertex.

4

The Hirsch Conjecture Counter-examples to Hirsch Simplicial complexes Normal complexes The Hirsch Conjecture

Motivation: LP

The feasibility region P = {x ∈ Rd : Ax ≤ b}, A ∈ Rn×d

• f a linear program is a polyhedron P with (at most) n

facets and d dimensions. Optimal value (if bounded) is always attained at a vertex. The simplex method (Dantzig 1947) solves the linear program by starting at any feasible vertex and moving along the graph of P, in a monotone fashion, until the

• ptimum is attained.

4

The Hirsch Conjecture Counter-examples to Hirsch Simplicial complexes Normal complexes The Hirsch Conjecture

Motivation: LP

The feasibility region P = {x ∈ Rd : Ax ≤ b}, A ∈ Rn×d

• f a linear program is a polyhedron P with (at most) n

facets and d dimensions. Optimal value (if bounded) is always attained at a vertex. The simplex method (Dantzig 1947) solves the linear program by starting at any feasible vertex and moving along the graph of P, in a monotone fashion, until the

• ptimum is attained.

In particular, a polynomial pivot rule for the simplex method would prove that Linear Programming can be performed in strongly polynomial time. (One of Smale’s "problems for the 21st century").

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The Hirsch Conjecture Counter-examples to Hirsch Simplicial complexes Normal complexes The Hirsch Conjecture

Polynomial Hirsch conjecture

In this sense, more important than the original Hirsch conjecture is the following “polynomial version” of it: Polynomial Hirsch Conjecture There is a constant k such that: H(d, n) ≤ nk, ∀n, d, where H(d, n) denotes the maximum diameter of d-polyhedra with n facets.

5

The Hirsch Conjecture Counter-examples to Hirsch Simplicial complexes Normal complexes General bounds and known cases

Two general bounds

Theorem (Kalai-Kleitman, 1992, “quasi-polynomial”) H(d, n) ≤ nlog2 d+2, ∀n, d. Theorem (Larman, 1970; Barnette, 1974, linear in fixed d) H(d, n) ≤ 2d−3n, ∀n, d.

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The Hirsch Conjecture Counter-examples to Hirsch Simplicial complexes Normal complexes General bounds and known cases

Two classical reductions: (1) simplicity

Theorem (Klee, 1964) For every n, d the maximum H(d, n) is attained at a simple polyhedron. P simple ⇔ Every vertex of P lies in exactly d-facets

7

The Hirsch Conjecture Counter-examples to Hirsch Simplicial complexes Normal complexes General bounds and known cases

Two classical reductions: (1) simplicity

Theorem (Klee, 1964) For every n, d the maximum H(d, n) is attained at a simple polyhedron. P simple ⇔ Every vertex of P lies in exactly d-facets ⇔ The (polar) dual of P is a pure simplicial complex with the topology of a (d − 1)-sphere (if P is bounded) or a (d − 1)-ball (if P is unbounded).

7

The Hirsch Conjecture Counter-examples to Hirsch Simplicial complexes Normal complexes General bounds and known cases

Two classical reductions: (1) simplicity

Theorem (Klee, 1964) For every n, d the maximum H(d, n) is attained at a simple polyhedron. P simple ⇔ Every vertex of P lies in exactly d-facets ⇔ The (polar) dual of P is a pure simplicial complex with the topology of a (d − 1)-sphere (if P is bounded) or a (d − 1)-ball (if P is unbounded). This suggests a purely combinatorial/topological approach to questions about the diameter: study the adjacency-graph diameter of pure simplicial complexes (perhaps with extra requirements).

7

The Hirsch Conjecture Counter-examples to Hirsch Simplicial complexes Normal complexes General bounds and known cases

Two classical reductions: (2) d-step theorem

Theorem (Klee-Walkup, 1967) For every n, d, H(d, n) ≤ H(n − d, 2n − 2d).

8

The Hirsch Conjecture Counter-examples to Hirsch Simplicial complexes Normal complexes General bounds and known cases

Two classical reductions: (2) d-step theorem

Theorem (Klee-Walkup, 1967) For every n, d, H(d, n) ≤ H(n − d, 2n − 2d). Corollary (d-step theorem) There is a function f(k) := H(k, 2k) such that H(d, n) ≤ f(n − d), ∀n, d.

8

The Hirsch Conjecture Counter-examples to Hirsch Simplicial complexes Normal complexes General bounds and known cases

Two classical reductions: (2) d-step theorem

Theorem (Klee-Walkup, 1967) For every n, d, H(d, n) ≤ H(n − d, 2n − 2d). Corollary (d-step theorem) There is a function f(k) := H(k, 2k) such that H(d, n) ≤ f(n − d), ∀n, d. In order to bound H(d, n) it suffices to look at H(k, 2k).

8

The Hirsch Conjecture Counter-examples to Hirsch Simplicial complexes Normal complexes

Counter-examples to the Hirsch Conjecture

The construction of counter-examples to the Hirsch conjecture has two ingredients:

9

The Hirsch Conjecture Counter-examples to Hirsch Simplicial complexes Normal complexes

Counter-examples to the Hirsch Conjecture

The construction of counter-examples to the Hirsch conjecture has two ingredients:

1

A strong d-step theorem for spindles/prismatoids.

9

The Hirsch Conjecture Counter-examples to Hirsch Simplicial complexes Normal complexes

Counter-examples to the Hirsch Conjecture

The construction of counter-examples to the Hirsch conjecture has two ingredients:

1

A strong d-step theorem for spindles/prismatoids.

2

The construction of a prismatoid of dimension 5 and “width” 6.

9

The Hirsch Conjecture Counter-examples to Hirsch Simplicial complexes Normal complexes The Strong d-step Theorem

The strong d-step Theorem

The Klee-Walkup d-step Theorem follows from the following lemma:

10

The Hirsch Conjecture Counter-examples to Hirsch Simplicial complexes Normal complexes The Strong d-step Theorem

The strong d-step Theorem

The Klee-Walkup d-step Theorem follows from the following lemma: Lemma (Klee-Walkup 1967) For every d-polytope P with n > 2d facets and diameter δ there is a d + 1-polytope with one more facet and the same diameter δ.

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The Hirsch Conjecture Counter-examples to Hirsch Simplicial complexes Normal complexes The Strong d-step Theorem

The strong d-step Theorem

The Klee-Walkup d-step Theorem follows from the following lemma: Lemma (Klee-Walkup 1967) For every d-polytope P with n > 2d facets and diameter δ there is a d + 1-polytope with one more facet and the same diameter δ. The strong d-step Theorem is the following modification of it: Lemma (S. 2012) For every d-spindle P with n > 2d facets and length λ there is a d + 1-spindle with one more facet and length λ + 1.

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The Hirsch Conjecture Counter-examples to Hirsch Simplicial complexes Normal complexes The Strong d-step Theorem

Spindles

Definition A spindle is a polytope P with two distinguished vertices u and v such that every facet contains either u or v (but not both).

u u v v

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The Hirsch Conjecture Counter-examples to Hirsch Simplicial complexes Normal complexes The Strong d-step Theorem

Spindles

Definition A spindle is a polytope P with two distinguished vertices u and v such that every facet contains either u or v (but not both).

u u v v

Definition The length of a spindle is the graph distance from u to v.

11

The Hirsch Conjecture Counter-examples to Hirsch Simplicial complexes Normal complexes The Strong d-step Theorem

Spindles

Definition A spindle is a polytope P with two distinguished vertices u and v such that every facet contains either u or v (but not both).

u u v v

Definition The length of a spindle is the graph distance from u to v. Exercise 3-spindles have length ≤ 3.

11

The Hirsch Conjecture Counter-examples to Hirsch Simplicial complexes Normal complexes The Strong d-step Theorem

Prismatoids

Definition A prismatoid is a polytope Q with two (parallel) facets Q+ and Q− containing all vertices.

Q+ Q− Q

12

The Hirsch Conjecture Counter-examples to Hirsch Simplicial complexes Normal complexes The Strong d-step Theorem

Prismatoids

Definition A prismatoid is a polytope Q with two (parallel) facets Q+ and Q− containing all vertices.

Q+ Q− Q

Definition The width of a prismatoid is the dual-graph distance from Q+ to Q−.

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The Hirsch Conjecture Counter-examples to Hirsch Simplicial complexes Normal complexes The Strong d-step Theorem

Prismatoids

Definition A prismatoid is a polytope Q with two (parallel) facets Q+ and Q− containing all vertices.

Q+ Q− Q

Definition The width of a prismatoid is the dual-graph distance from Q+ to Q−. Exercise 3-prismatoids have width ≤ 3.

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The Hirsch Conjecture Counter-examples to Hirsch Simplicial complexes Normal complexes The Strong d-step Theorem

Prismatoids

Theorem (Strong d-step theorem, prismatoid version) Let Q be a prismatoid of dimension d, with n > 2d vertices and width δ. Then there is another prismatoid Q′ of dimension d + 1, with n + 1 vertices and width δ + 1.

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The Hirsch Conjecture Counter-examples to Hirsch Simplicial complexes Normal complexes The Strong d-step Theorem

Prismatoids

Theorem (Strong d-step theorem, prismatoid version) Let Q be a prismatoid of dimension d, with n > 2d vertices and width δ. Then there is another prismatoid Q′ of dimension d + 1, with n + 1 vertices and width δ + 1. That is: we can increase the dimension, width and number of vertices of a prismatoid, all by one, until n = 2d.

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The Hirsch Conjecture Counter-examples to Hirsch Simplicial complexes Normal complexes The Strong d-step Theorem

Prismatoids

Theorem (Strong d-step theorem, prismatoid version) Let Q be a prismatoid of dimension d, with n > 2d vertices and width δ. Then there is another prismatoid Q′ of dimension d + 1, with n + 1 vertices and width δ + 1. That is: we can increase the dimension, width and number of vertices of a prismatoid, all by one, until n = 2d. Corollary In particular, if a prismatoid Q has width > d then there is another prismatoid Q′ (of dimension n − d, with 2n − 2d vertices, and

width ≥ δ + n − 2d > n − d) that violates (the dual of) the Hirsch

conjecture.

13

The Hirsch Conjecture Counter-examples to Hirsch Simplicial complexes Normal complexes The Strong d-step Theorem

d-step theorem for prismatoids

Proof.

Q ⊂ R2 Q+ Q−

• Q−
• Q ⊂ R3
• Q+

w

• Q− := opsv(Q−)

Q+ w

• psv(Q) ⊂ R3

v u u

14

The Hirsch Conjecture Counter-examples to Hirsch Simplicial complexes Normal complexes The Strong d-step Theorem

Width of prismatoids

So, to disprove the Hirsch Conjecture we only need to find a prismatoid of dimension d and width larger than d.

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The Hirsch Conjecture Counter-examples to Hirsch Simplicial complexes Normal complexes The Strong d-step Theorem

Width of prismatoids

So, to disprove the Hirsch Conjecture we only need to find a prismatoid of dimension d and width larger than d. Its number

• f vertices and facets is irrelevant...

15

The Hirsch Conjecture Counter-examples to Hirsch Simplicial complexes Normal complexes The Strong d-step Theorem

Width of prismatoids

So, to disprove the Hirsch Conjecture we only need to find a prismatoid of dimension d and width larger than d. Its number

• f vertices and facets is irrelevant...

Question Do they exist?

15

The Hirsch Conjecture Counter-examples to Hirsch Simplicial complexes Normal complexes The Strong d-step Theorem

Width of prismatoids

So, to disprove the Hirsch Conjecture we only need to find a prismatoid of dimension d and width larger than d. Its number

• f vertices and facets is irrelevant...

Question Do they exist? 3-prismatoids have width at most 3 (exercise).

15

The Hirsch Conjecture Counter-examples to Hirsch Simplicial complexes Normal complexes The Strong d-step Theorem

Width of prismatoids

So, to disprove the Hirsch Conjecture we only need to find a prismatoid of dimension d and width larger than d. Its number

• f vertices and facets is irrelevant...

Question Do they exist? 3-prismatoids have width at most 3 (exercise). 4-prismatoids have width at most 4 [S.-Stephen-Thomas, 2012].

15

The Hirsch Conjecture Counter-examples to Hirsch Simplicial complexes Normal complexes The Strong d-step Theorem

Width of prismatoids

So, to disprove the Hirsch Conjecture we only need to find a prismatoid of dimension d and width larger than d. Its number

• f vertices and facets is irrelevant...

Question Do they exist? 3-prismatoids have width at most 3 (exercise). 4-prismatoids have width at most 4 [S.-Stephen-Thomas, 2012]. 5-prismatoids of width 6 exist [S., 2012]

15

The Hirsch Conjecture Counter-examples to Hirsch Simplicial complexes Normal complexes The Strong d-step Theorem

Width of prismatoids

So, to disprove the Hirsch Conjecture we only need to find a prismatoid of dimension d and width larger than d. Its number

• f vertices and facets is irrelevant...

Question Do they exist? 3-prismatoids have width at most 3 (exercise). 4-prismatoids have width at most 4 [S.-Stephen-Thomas, 2012]. 5-prismatoids of width 6 exist [S., 2012] with 25 vertices [Matschke-S.-Weibel 2015].

15

The Hirsch Conjecture Counter-examples to Hirsch Simplicial complexes Normal complexes The Strong d-step Theorem

Width of prismatoids

So, to disprove the Hirsch Conjecture we only need to find a prismatoid of dimension d and width larger than d. Its number

• f vertices and facets is irrelevant...

Question Do they exist? 3-prismatoids have width at most 3 (exercise). 4-prismatoids have width at most 4 [S.-Stephen-Thomas, 2012]. 5-prismatoids of width 6 exist [S., 2012] with 25 vertices [Matschke-S.-Weibel 2015]. 5-prismatoids of arbitrarily large width exist [Matschke-S.-Weibel 2015].

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The Hirsch Conjecture Counter-examples to Hirsch Simplicial complexes Normal complexes Wide prismatoids

Combinatorics of prismatoids

Analyzing the combinatorics of a d-prismatoid Q can be done via an intermediate slice . . .

Q+ Q− Q ∩ H H Q

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The Hirsch Conjecture Counter-examples to Hirsch Simplicial complexes Normal complexes Wide prismatoids

Combinatorics of prismatoids

. . . which equals the (averaged) Minkowski sum Q+ + Q− of the two bases Q+ and Q−. The normal fan of Q+ + Q− equals the “superposition” of those of Q+ and Q−.

+ 1

2 1 2

=

16

The Hirsch Conjecture Counter-examples to Hirsch Simplicial complexes Normal complexes Wide prismatoids

Combinatorics of prismatoids

. . . which equals the (averaged) Minkowski sum Q+ + Q− of the two bases Q+ and Q−. The normal fan of Q+ + Q− equals the “superposition” of those of Q+ and Q−.

+ 1

2 1 2

=

16

The Hirsch Conjecture Counter-examples to Hirsch Simplicial complexes Normal complexes Wide prismatoids

Combinatorics of prismatoids

So: the combinatorics of Q follows from the superposition of the normal fans of Q+ and Q−.

17

The Hirsch Conjecture Counter-examples to Hirsch Simplicial complexes Normal complexes Wide prismatoids

Combinatorics of prismatoids

So: the combinatorics of Q follows from the superposition of the normal fans of Q+ and Q−. Remark The normal fan of a d − 1-polytope can be thought of as a (geodesic, polytopal) cell decomposition (“a map”) in the d − 2-sphere.

17

The Hirsch Conjecture Counter-examples to Hirsch Simplicial complexes Normal complexes Wide prismatoids

Combinatorics of prismatoids

So: the combinatorics of Q follows from the superposition of the normal fans of Q+ and Q−. Remark The normal fan of a d − 1-polytope can be thought of as a (geodesic, polytopal) cell decomposition (“a map”) in the d − 2-sphere. That is: in order to construct and understand prismatoids of dimension 5 you only need to look at pairs of geodesic cell decompositions of the 3-sphere.

17

The Hirsch Conjecture Counter-examples to Hirsch Simplicial complexes Normal complexes Wide prismatoids

Example: a 3-prismatoid

+ 1

2 1 2

=

18

The Hirsch Conjecture Counter-examples to Hirsch Simplicial complexes Normal complexes Wide 5-prismatoids

A 5-prismatoid of width > 5

Theorem (S. 2012) The following prismatoid Q, of dimension 5 and with 48 vertices, has width six.

Q := conv                                           x1 x2 x3 x4 x5 ±18 1 ±18 1 ±45 1 ±45 1 ±15 ±15 1 ±30 ±30 1 ±10 ±40 1 ±10 ±40 1                       x1 x2 x3 x4 x5 ±18 −1 ±18 −1 ±45 −1 ±45 −1 ±15 ±15 −1 ±30 ±30 −1 ±40 ±10 −1 ±40 ±10 −1                                           19

The Hirsch Conjecture Counter-examples to Hirsch Simplicial complexes Normal complexes Wide 5-prismatoids

A 5-prismatoid of width > 5

Theorem (S. 2012) The following prismatoid Q, of dimension 5 and with 48 vertices, has width six.

Q := conv                                           x1 x2 x3 x4 x5 ±18 1 ±18 1 ±45 1 ±45 1 ±15 ±15 1 ±30 ±30 1 ±10 ±40 1 ±10 ±40 1                       x1 x2 x3 x4 x5 ±18 −1 ±18 −1 ±45 −1 ±45 −1 ±15 ±15 −1 ±30 ±30 −1 ±40 ±10 −1 ±40 ±10 −1                                          

Corollary There is a 43-dimensional polytope with 86 facets and diameter (at least) 44.

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The Hirsch Conjecture Counter-examples to Hirsch Simplicial complexes Normal complexes Wide 5-prismatoids

Smaller 5-prismatoids of width > 5

And with some more work: Theorem (Matschke-Santos-Weibel, 2015) There is a 5-prismatoid with 25 vertices and of width 6. Corollary There is a non-Hirsch polytope of dimension 20 with 40 facets.

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The Hirsch Conjecture Counter-examples to Hirsch Simplicial complexes Normal complexes Wide 5-prismatoids

Smaller 5-prismatoids of width > 5

And with some more work: Theorem (Matschke-Santos-Weibel, 2015) There is a 5-prismatoid with 25 vertices and of width 6. Corollary There is a non-Hirsch polytope of dimension 20 with 40 facets. This one has been explicitly computed. It has 36, 442 vertices, and diameter 21.

20

The Hirsch Conjecture Counter-examples to Hirsch Simplicial complexes Normal complexes Wide 5-prismatoids 21

The Hirsch Conjecture Counter-examples to Hirsch Simplicial complexes Normal complexes How far can we go

Many non-Hirsch polytopes

Once we have a non-Hirsch polytope we can derive more via:

1

Products of several copies of it (dimension increases).

2

Gluing (or, rather, “blending”) several copies of it (dimension is fixed).

22

The Hirsch Conjecture Counter-examples to Hirsch Simplicial complexes Normal complexes How far can we go

Many non-Hirsch polytopes

Once we have a non-Hirsch polytope we can derive more via:

1

Products of several copies of it (dimension increases).

2

Gluing (or, rather, “blending”) several copies of it (dimension is fixed). To analyze the asymptotics of these operations, we call Hirsch excess of a d-polytope P with n facets and diameter δ the number ǫ(P) := δ n − d − 1 = δ − (n − d) n − d .

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The Hirsch Conjecture Counter-examples to Hirsch Simplicial complexes Normal complexes How far can we go

Many non-Hirsch polytopes

Once we have a non-Hirsch polytope we can derive more via:

1

Products of several copies of it (dimension increases).

2

Gluing (or, rather, “blending”) several copies of it (dimension is fixed). To analyze the asymptotics of these operations, we call Hirsch excess of a d-polytope P with n facets and diameter δ the number ǫ(P) := δ n − d − 1 = δ − (n − d) n − d .

• E. g.: The excess of our non-Hirsch polytope with n − d = 20

and with diameter 21 is 21 − 20 20 = 5%.

22

The Hirsch Conjecture Counter-examples to Hirsch Simplicial complexes Normal complexes How far can we go

Many non-Hirsch polytopes

1

Taking products preserves the excess. Corollary For each k ∈ N there is a non-Hirsch polytope of dimension 20k with 40k facets and with excess 0.05.

23

The Hirsch Conjecture Counter-examples to Hirsch Simplicial complexes Normal complexes How far can we go

Many non-Hirsch polytopes

1

Taking products preserves the excess. Corollary For each k ∈ N there is a non-Hirsch polytope of dimension 20k with 40k facets and with excess 0.05.

2

Gluing several copies (slightly) decreases the excess.

23

The Hirsch Conjecture Counter-examples to Hirsch Simplicial complexes Normal complexes How far can we go

Many non-Hirsch polytopes

1

Taking products preserves the excess. Corollary For each k ∈ N there is a non-Hirsch polytope of dimension 20k with 40k facets and with excess 0.05.

2

Gluing several copies (slightly) decreases the excess. Corollary For each k ∈ N there is an infinite family of non-Hirsch polytopes of fixed dimension 20k and with excess (tending to) 0.05

• 1 − 1

k

• .

23

The Hirsch Conjecture Counter-examples to Hirsch Simplicial complexes Normal complexes General simplicial complexes

If you cannot prove it, generalize it

We know that H(d, n) is attained at a simple polytope, whose polar is a simplicial polytope.

24

The Hirsch Conjecture Counter-examples to Hirsch Simplicial complexes Normal complexes General simplicial complexes

If you cannot prove it, generalize it

We know that H(d, n) is attained at a simple polytope, whose polar is a simplicial polytope. Instead of looking only at (simplicial) polytopes, why not look at the maximum diameter of more general simplicial complexes?

24

The Hirsch Conjecture Counter-examples to Hirsch Simplicial complexes Normal complexes General simplicial complexes

If you cannot prove it, generalize it

We know that H(d, n) is attained at a simple polytope, whose polar is a simplicial polytope. Instead of looking only at (simplicial) polytopes, why not look at the maximum diameter of more general simplicial complexes? Definition A pure simplicial complex K of dimension d − 1 with n vertices is a subset of [n]

d

• . That is, a family of size d subsets of [n] := {1, . . . , n}.

The elements of K are called facets. Any subset of a facet is a face of K.

24

The Hirsch Conjecture Counter-examples to Hirsch Simplicial complexes Normal complexes General simplicial complexes

If you cannot prove it, generalize it

We know that H(d, n) is attained at a simple polytope, whose polar is a simplicial polytope. Instead of looking only at (simplicial) polytopes, why not look at the maximum diameter of more general simplicial complexes? Definition A pure simplicial complex K of dimension d − 1 with n vertices is a subset of [n]

d

• . That is, a family of size d subsets of [n] := {1, . . . , n}.

The elements of K are called facets. Any subset of a facet is a face of K. The (adjacency) graph of K has its d-sets (a.k.a. facets) as nodes, and two facets X, Y are adjacent if X ∩ Y = d − 1.

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The Hirsch Conjecture Counter-examples to Hirsch Simplicial complexes Normal complexes General simplicial complexes

Several versions of the question:

25

The Hirsch Conjecture Counter-examples to Hirsch Simplicial complexes Normal complexes General simplicial complexes

Several versions of the question:

Pure simplicial complexes, in general. Hc(d, n)

25

The Hirsch Conjecture Counter-examples to Hirsch Simplicial complexes Normal complexes General simplicial complexes

Several versions of the question:

Pure simplicial complexes, in general. Hc(d, n) Pseudo-manifolds (w. or wo. bdry). Hpm(d, n), Hpm(d, n)

25

The Hirsch Conjecture Counter-examples to Hirsch Simplicial complexes Normal complexes General simplicial complexes

Several versions of the question:

Pure simplicial complexes, in general. Hc(d, n) Pseudo-manifolds (w. or wo. bdry). Hpm(d, n), Hpm(d, n) Manifolds (w. or wo. bdry). HM(d, n), HM(d, n)

25

The Hirsch Conjecture Counter-examples to Hirsch Simplicial complexes Normal complexes General simplicial complexes

Several versions of the question:

Pure simplicial complexes, in general. Hc(d, n) Pseudo-manifolds (w. or wo. bdry). Hpm(d, n), Hpm(d, n) Manifolds (w. or wo. bdry). HM(d, n), HM(d, n) Spheres (or balls). HS(d, n), HB(d, n), . . .

25

The Hirsch Conjecture Counter-examples to Hirsch Simplicial complexes Normal complexes General simplicial complexes

Several versions of the question:

Pure simplicial complexes, in general. Hc(d, n) Pseudo-manifolds (w. or wo. bdry). Hpm(d, n), Hpm(d, n) Manifolds (w. or wo. bdry). HM(d, n), HM(d, n) Spheres (or balls). HS(d, n), HB(d, n), . . . H∗(d, n) is the maximum (dual) diameter; two simplices are considered adjacent if they differ by a single vertex.

25

The Hirsch Conjecture Counter-examples to Hirsch Simplicial complexes Normal complexes General simplicial complexes

Several versions of the question:

Pure simplicial complexes, in general. Hc(d, n) Pseudo-manifolds (w. or wo. bdry). Hpm(d, n), Hpm(d, n) Manifolds (w. or wo. bdry). HM(d, n), HM(d, n) Spheres (or balls). HS(d, n), HB(d, n), . . . H∗(d, n) is the maximum (dual) diameter; two simplices are considered adjacent if they differ by a single vertex. Lemma Hc(d, n) is attained at a complex whose (dual) graph is a path (in particular, at a pseudo-manifold w. bdry.) for every n, d.

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The Hirsch Conjecture Counter-examples to Hirsch Simplicial complexes Normal complexes General simplicial complexes

The maximum diameter of pure simplicial complexes

In dimension two: Theorem (S., 2013) 2 9(n − 1)2 < Hc(3, n) < 1 4n2.

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The Hirsch Conjecture Counter-examples to Hirsch Simplicial complexes Normal complexes General simplicial complexes

The maximum diameter of pure simplicial complexes

In dimension two: Theorem (S., 2013) 2 9(n − 1)2 < Hc(3, n) < 1 4n2. In higher dimension: Theorem (S., 2013) Hc(md, mn) > 2 2m Hc(d, n)m, ∀m ∈ N.

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The Hirsch Conjecture Counter-examples to Hirsch Simplicial complexes Normal complexes General simplicial complexes

The maximum diameter of pure simplicial complexes

In dimension two: Theorem (S., 2013) 2 9(n − 1)2 < Hc(3, n) < 1 4n2. In higher dimension: Theorem (S., 2013) Hc(md, mn) > 2 2m Hc(d, n)m, ∀m ∈ N. Corollary (S., 2013) Ω(n2d/3) ≤ Hc(d, n) ≤ O(nd).

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The Hirsch Conjecture Counter-examples to Hirsch Simplicial complexes Normal complexes General simplicial complexes

Hc(3, n) > 2

9(n − 1)2

Proof:

1

Without loss of generality assume n = 3k + 1.

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The Hirsch Conjecture Counter-examples to Hirsch Simplicial complexes Normal complexes General simplicial complexes

Hc(3, n) > 2

9(n − 1)2

Proof:

1

Without loss of generality assume n = 3k + 1.

2

With the first 2k + 1 vertices, construct k disjoint cycles of length 2k + 1

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The Hirsch Conjecture Counter-examples to Hirsch Simplicial complexes Normal complexes General simplicial complexes

Hc(3, n) > 2

9(n − 1)2

Proof:

1

Without loss of generality assume n = 3k + 1.

2

With the first 2k + 1 vertices, construct k disjoint cycles of length 2k + 1 (That is, decompose K2k+1 into k disjoint Hamiltonian cycles).

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The Hirsch Conjecture Counter-examples to Hirsch Simplicial complexes Normal complexes General simplicial complexes

Hc(3, n) > 2

9(n − 1)2

Proof:

1

Without loss of generality assume n = 3k + 1.

2

With the first 2k + 1 vertices, construct k disjoint cycles of length 2k + 1 (That is, decompose K2k+1 into k disjoint Hamiltonian cycles).

3

Remove an edge from each cycle to make it a path, and join each path to each of the remaining k vertices.

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The Hirsch Conjecture Counter-examples to Hirsch Simplicial complexes Normal complexes General simplicial complexes

Hc(3, n) > 2

9(n − 1)2

Proof:

1

Without loss of generality assume n = 3k + 1.

2

With the first 2k + 1 vertices, construct k disjoint cycles of length 2k + 1 (That is, decompose K2k+1 into k disjoint Hamiltonian cycles).

3

Remove an edge from each cycle to make it a path, and join each path to each of the remaining k vertices.

4

Glue together the k chains using k − 1 triangles.

27

The Hirsch Conjecture Counter-examples to Hirsch Simplicial complexes Normal complexes General simplicial complexes

Hc(3, n) > 2

9(n − 1)2

Proof:

1

Without loss of generality assume n = 3k + 1.

2

With the first 2k + 1 vertices, construct k disjoint cycles of length 2k + 1 (That is, decompose K2k+1 into k disjoint Hamiltonian cycles).

3

Remove an edge from each cycle to make it a path, and join each path to each of the remaining k vertices.

4

Glue together the k chains using k − 1 triangles. In this way we get a chain of triangles of length (2k + 1)k − 2 > 2 9(3k)2.

27

The Hirsch Conjecture Counter-examples to Hirsch Simplicial complexes Normal complexes General simplicial complexes

Hc(3, n) > 2

9(n − 1)2

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The Hirsch Conjecture Counter-examples to Hirsch Simplicial complexes Normal complexes General simplicial complexes

Hc(md, mn) > 1

2Hc(d, n)m

Proof:

1

Let K be a complex achieving HC(d, n). W.l.o.g. assume its dual graph is a path.

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The Hirsch Conjecture Counter-examples to Hirsch Simplicial complexes Normal complexes General simplicial complexes

Hc(md, mn) > 1

2Hc(d, n)m

Proof:

1

Let K be a complex achieving HC(d, n). W.l.o.g. assume its dual graph is a path.

2

Take the join K ∗m := K ∗ K ∗ · · · ∗ K of m copies of K.

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The Hirsch Conjecture Counter-examples to Hirsch Simplicial complexes Normal complexes General simplicial complexes

Hc(md, mn) > 1

2Hc(d, n)m

Proof:

1

Let K be a complex achieving HC(d, n). W.l.o.g. assume its dual graph is a path.

2

Take the join K ∗m := K ∗ K ∗ · · · ∗ K of m copies of K. K ∗m is a complex of dimension md − 1, with mn vertices and whose dual graph is an m-dimensional grid of side HC(d, n). (It has (HC(d, n) + 1)m facets).

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The Hirsch Conjecture Counter-examples to Hirsch Simplicial complexes Normal complexes General simplicial complexes

Hc(md, mn) > 1

2Hc(d, n)m

Proof:

1

Let K be a complex achieving HC(d, n). W.l.o.g. assume its dual graph is a path.

2

Take the join K ∗m := K ∗ K ∗ · · · ∗ K of m copies of K. K ∗m is a complex of dimension md − 1, with mn vertices and whose dual graph is an m-dimensional grid of side HC(d, n). (It has (HC(d, n) + 1)m facets).

3

In this grid consider a maximal induced path. This can be done using more than half of the vertices.

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The Hirsch Conjecture Counter-examples to Hirsch Simplicial complexes Normal complexes General simplicial complexes

Hc(md, mn) > 1

2Hc(d, n)m

30

The Hirsch Conjecture Counter-examples to Hirsch Simplicial complexes Normal complexes Normal complexes

So, pure simplicial complexes (even pseudo-manifolds) can have exponential diameters. What restriction should we put for (having at least hopes of) getting polynomial diameters?

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The Hirsch Conjecture Counter-examples to Hirsch Simplicial complexes Normal complexes Normal complexes

So, pure simplicial complexes (even pseudo-manifolds) can have exponential diameters. What restriction should we put for (having at least hopes of) getting polynomial diameters? It seems that everybody’s favorite is: Definition A simplicial complex K is called normal or locally strongly connected if, for every face S ∈ K, the adjacency subgraph induced by facets containing S is connected.

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The Hirsch Conjecture Counter-examples to Hirsch Simplicial complexes Normal complexes Normal complexes

So, pure simplicial complexes (even pseudo-manifolds) can have exponential diameters. What restriction should we put for (having at least hopes of) getting polynomial diameters? It seems that everybody’s favorite is: Definition A simplicial complex K is called normal or locally strongly connected if, for every face S ∈ K, the adjacency subgraph induced by facets containing S is connected. That is, if between every two facets X, Y there is a path using

• nly facets that contain X ∩ Y.

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The Hirsch Conjecture Counter-examples to Hirsch Simplicial complexes Normal complexes Normal complexes

So, pure simplicial complexes (even pseudo-manifolds) can have exponential diameters. What restriction should we put for (having at least hopes of) getting polynomial diameters? It seems that everybody’s favorite is: Definition A simplicial complex K is called normal or locally strongly connected if, for every face S ∈ K, the adjacency subgraph induced by facets containing S is connected. That is, if between every two facets X, Y there is a path using

• nly facets that contain X ∩ Y.

Adler and Dantzig (1974) call a normal pseudo-manifold an abstract polytope

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The Hirsch Conjecture Counter-examples to Hirsch Simplicial complexes Normal complexes Normal complexes

The importance of being normal

Normality is a hereditary property. Every link in a normal complex is normal, which is convenient for proofs by induction on d.

32

The Hirsch Conjecture Counter-examples to Hirsch Simplicial complexes Normal complexes Normal complexes

The importance of being normal

Normality is a hereditary property. Every link in a normal complex is normal, which is convenient for proofs by induction on d. One can argue that the dual graph of a complex only captures proximity if the complex is normal.

32

The Hirsch Conjecture Counter-examples to Hirsch Simplicial complexes Normal complexes Normal complexes

The importance of being normal

Normality is a hereditary property. Every link in a normal complex is normal, which is convenient for proofs by induction on d. One can argue that the dual graph of a complex only captures proximity if the complex is normal. Manifolds (w. or wo. boundary) are normal, but pseudo-manifolds are, in general, not.

32

The Hirsch Conjecture Counter-examples to Hirsch Simplicial complexes Normal complexes Normal complexes

The importance of being normal

Normality is a hereditary property. Every link in a normal complex is normal, which is convenient for proofs by induction on d. One can argue that the dual graph of a complex only captures proximity if the complex is normal. Manifolds (w. or wo. boundary) are normal, but pseudo-manifolds are, in general, not. The bounds we have for polytopes work for all normal complexes:

32

The Hirsch Conjecture Counter-examples to Hirsch Simplicial complexes Normal complexes Normal complexes

The importance of being normal

Normality is a hereditary property. Every link in a normal complex is normal, which is convenient for proofs by induction on d. One can argue that the dual graph of a complex only captures proximity if the complex is normal. Manifolds (w. or wo. boundary) are normal, but pseudo-manifolds are, in general, not. The bounds we have for polytopes work for all normal complexes: Theorem (Kalai-Kleitman 1992, Larman 1970) The (dual) diameter of every pure normal d-complex with n vertices is bounded above by Hn(d, n) ≤ nlog d+1, Hn(d, n) ≤ 2d−2n.

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The Hirsch Conjecture Counter-examples to Hirsch Simplicial complexes Normal complexes Normal complexes

Combinatorial segments in normal complexes

Inspired by Larman’s proof, Adiprasito-Benedetti (2014) define a combinatorial segment in a simplicial complex K to be any adjacency path with certain particular properties:

33

The Hirsch Conjecture Counter-examples to Hirsch Simplicial complexes Normal complexes Normal complexes

Combinatorial segments in normal complexes

Inspired by Larman’s proof, Adiprasito-Benedetti (2014) define a combinatorial segment in a simplicial complex K to be any adjacency path with certain particular properties: The path tries to get to a target set S of vertices but is anchored at a certain vertex v, which is only abandoned when a vertex v′ closer to S is found.

33

The Hirsch Conjecture Counter-examples to Hirsch Simplicial complexes Normal complexes Normal complexes

Combinatorial segments in normal complexes

Inspired by Larman’s proof, Adiprasito-Benedetti (2014) define a combinatorial segment in a simplicial complex K to be any adjacency path with certain particular properties: The path tries to get to a target set S of vertices but is anchored at a certain vertex v, which is only abandoned when a vertex v′ closer to S is found. While anchored at v, the path is a combinatorial segment in the star of v with target set Sv := neighbors of v that are closer to S than v

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The Hirsch Conjecture Counter-examples to Hirsch Simplicial complexes Normal complexes Normal complexes

Properties of combinatorial segments

Theorem (Adiprasito-Benedetti 2014) Let X, Y be facets in a normal simplicial complex K of dimension d − 1 with n vertices. Then:

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The Hirsch Conjecture Counter-examples to Hirsch Simplicial complexes Normal complexes Normal complexes

Properties of combinatorial segments

Theorem (Adiprasito-Benedetti 2014) Let X, Y be facets in a normal simplicial complex K of dimension d − 1 with n vertices. Then:

1

There is at least one comb. segment from X to Y.

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The Hirsch Conjecture Counter-examples to Hirsch Simplicial complexes Normal complexes Normal complexes

Properties of combinatorial segments

Theorem (Adiprasito-Benedetti 2014) Let X, Y be facets in a normal simplicial complex K of dimension d − 1 with n vertices. Then:

1

There is at least one comb. segment from X to Y.

2

No combinatorial segment is longer then n2d−2.

34

The Hirsch Conjecture Counter-examples to Hirsch Simplicial complexes Normal complexes Normal complexes

Properties of combinatorial segments

Theorem (Adiprasito-Benedetti 2014) Let X, Y be facets in a normal simplicial complex K of dimension d − 1 with n vertices. Then:

1

There is at least one comb. segment from X to Y.

2

No combinatorial segment is longer then n2d−2.

3

If K is a flag simplicial complex then every combinatorial segment is of length ≤ n − d.

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The Hirsch Conjecture Counter-examples to Hirsch Simplicial complexes Normal complexes Normal complexes

Properties of combinatorial segments

Theorem (Adiprasito-Benedetti 2014) Let X, Y be facets in a normal simplicial complex K of dimension d − 1 with n vertices. Then:

1

There is at least one comb. segment from X to Y.

2

No combinatorial segment is longer then n2d−2.

3

If K is a flag simplicial complex then every combinatorial segment is of length ≤ n − d.

A simplicial complex is flag if it is the clique complex of its 1-skeleton. Equivalently, the minimal subsets of [n] that are not faces have size two.

34

The Hirsch Conjecture Counter-examples to Hirsch Simplicial complexes Normal complexes Normal complexes

Properties of combinatorial segments

Theorem (Adiprasito-Benedetti 2014) Let X, Y be facets in a normal simplicial complex K of dimension d − 1 with n vertices. Then:

1

There is at least one comb. segment from X to Y.

2

No combinatorial segment is longer then n2d−2.

3

If K is a flag simplicial complex then every combinatorial segment is of length ≤ n − d.

A simplicial complex is flag if it is the clique complex of its 1-skeleton. Equivalently, the minimal subsets of [n] that are not faces have size two.

Corollary Larman bound for normal simplicial complexes.

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The Hirsch Conjecture Counter-examples to Hirsch Simplicial complexes Normal complexes Normal complexes

Properties of combinatorial segments

Theorem (Adiprasito-Benedetti 2014) Let X, Y be facets in a normal simplicial complex K of dimension d − 1 with n vertices. Then:

1

There is at least one comb. segment from X to Y.

2

No combinatorial segment is longer then n2d−2.

3

If K is a flag simplicial complex then every combinatorial segment is of length ≤ n − d.

A simplicial complex is flag if it is the clique complex of its 1-skeleton. Equivalently, the minimal subsets of [n] that are not faces have size two.

Corollary Larman bound for normal simplicial complexes. Hirsch bound for all flag and normal simplicial complexes.

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The Hirsch Conjecture Counter-examples to Hirsch Simplicial complexes Normal complexes Normal complexes

Long combinatorial segments

35

The Hirsch Conjecture Counter-examples to Hirsch Simplicial complexes Normal complexes Normal complexes

Long combinatorial segments

So, the Larman bound not only bounds the diameter of a polytope or of a normal complex but also the length of any combinatorial segment in it.

35

The Hirsch Conjecture Counter-examples to Hirsch Simplicial complexes Normal complexes Normal complexes

Long combinatorial segments

So, the Larman bound not only bounds the diameter of a polytope or of a normal complex but also the length of any combinatorial segment in it. If we believe in the polynomial Hirsch Conjecture, could it be that all combinatorial segments have polynomial length?

35

The Hirsch Conjecture Counter-examples to Hirsch Simplicial complexes Normal complexes Normal complexes

Long combinatorial segments

So, the Larman bound not only bounds the diameter of a polytope or of a normal complex but also the length of any combinatorial segment in it. If we believe in the polynomial Hirsch Conjecture, could it be that all combinatorial segments have polynomial length? NO Theorem (Labbé-Manneville-S, 2016+) There are normal complexes having combinatorial segments of length n2d−2−ǫ, for any ǫ > 0.

35

The Hirsch Conjecture Counter-examples to Hirsch Simplicial complexes Normal complexes Normal complexes

Long combinatorial segments

So, the Larman bound not only bounds the diameter of a polytope or of a normal complex but also the length of any combinatorial segment in it. If we believe in the polynomial Hirsch Conjecture, could it be that all combinatorial segments have polynomial length? NO Theorem (Labbé-Manneville-S, 2016+) There are normal complexes having combinatorial segments of length n2d−2−ǫ, for any ǫ > 0. Theorem (Labbé-Manneville-S, 2016+) There are vertex-decomposable polytopes having combinatorial segments of length n2d−3−ǫ, for any ǫ > 0.

35

The Hirsch Conjecture Counter-examples to Hirsch Simplicial complexes Normal complexes Normal complexes

Long combinatorial segments

z1 F1 zN−1 x1

1

x2

1

x1

1

x2

1

x2 F2

36

The Hirsch Conjecture Counter-examples to Hirsch Simplicial complexes Normal complexes

T H A N K Y O U

37