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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 Matemticas, Estadstica y Computacin Universidad de

  1. 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: A strong d -step theorem for spindles/prismatoids. 1 9

  2. 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: A strong d -step theorem for spindles/prismatoids. 1 The construction of a prismatoid of dimension 5 and 2 “width” 6. 9

  3. 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

  4. 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 > 2 d facets and diameter δ there is a d + 1 -polytope with one more facet and the same diameter δ . 10

  5. 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 > 2 d 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 > 2 d facets and length λ there is a d + 1 -spindle with one more facet and length λ + 1 . 10

  6. 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). v v u u 11

  7. 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). Definition v v The length of a spindle is the graph distance from u to v . u u 11

  8. 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). Definition v v The length of a spindle is the graph distance from u to v . Exercise 3-spindles have length ≤ 3. u u 11

  9. 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

  10. 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. Definition The width of a prismatoid is the Q + dual-graph Q distance from Q + to Q − . Q − 12

  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. Definition The width of a prismatoid is the Q + dual-graph Q distance from Q + to Q − . Q − Exercise 3-prismatoids have width ≤ 3. 12

  12. 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 > 2 d vertices and width δ . Then there is another prismatoid Q ′ of dimension d + 1 , with n + 1 vertices and width δ + 1 . 13

  13. 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 > 2 d 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 = 2 d . 13

  14. 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 > 2 d 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 = 2 d . Corollary In particular, if a prismatoid Q has width > d then there is another prismatoid Q ′ (of dimension n − d, with 2 n − 2 d vertices, and width ≥ δ + n − 2 d > n − d) that violates (the dual of) the Hirsch conjecture. 13

  15. The Hirsch Conjecture Counter-examples to Hirsch Simplicial complexes Normal complexes The Strong d -step Theorem d -step theorem for prismatoids Proof. w v Q − � u Q − Q + � Q + Q ⊂ R 2 w � Q ⊂ R 3 Q − := ops v ( Q − ) � u Q + ops v ( Q ) ⊂ R 3 14

  16. 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 . 15

  17. 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 of vertices and facets is irrelevant... 15

  18. 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 of vertices and facets is irrelevant... Question Do they exist? 15

  19. 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 of vertices and facets is irrelevant... Question Do they exist? 3-prismatoids have width at most 3 (exercise). 15

  20. 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 of 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

  21. 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 of 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

  22. 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 of 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

  23. 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 of 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]. 15

  24. 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 − 16

  25. 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 + 1 = 2 2 16

  26. 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 + 1 = 2 2 16

  27. 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

  28. 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

  29. 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

  30. The Hirsch Conjecture Counter-examples to Hirsch Simplicial complexes Normal complexes Wide prismatoids Example: a 3-prismatoid 1 + 1 = 2 2 18

  31. 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.  x 1 x 2 x 3 x 4 x 5 x 1 x 2 x 3 x 4 x 5        ± 18 0 0 0 1   0 0 0 ± 18 − 1          0 ± 18 0 0 1 0 0 ± 18 0 − 1              0 0 ± 45 0 1   ± 45 0 0 0 − 1               0 0 0 ± 45 1   0 ± 45 0 0 − 1   Q := conv     ± 15 ± 15 0 0 1 0 0 ± 15 ± 15 − 1            0 0 ± 30 ± 30 1 ± 30 ± 30 0 0 − 1                 0 ± 10 ± 40 0 1 ± 40 0 ± 10 0 − 1             ± 10 0 0 ± 40 1 0 ± 40 0 ± 10 − 1            19

  32. 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.  x 1 x 2 x 3 x 4 x 5 x 1 x 2 x 3 x 4 x 5        ± 18 0 0 0 1   0 0 0 ± 18 − 1          0 ± 18 0 0 1 0 0 ± 18 0 − 1              0 0 ± 45 0 1   ± 45 0 0 0 − 1               0 0 0 ± 45 1   0 ± 45 0 0 − 1   Q := conv     ± 15 ± 15 0 0 1 0 0 ± 15 ± 15 − 1            0 0 ± 30 ± 30 1 ± 30 ± 30 0 0 − 1                 0 ± 10 ± 40 0 1 ± 40 0 ± 10 0 − 1             ± 10 0 0 ± 40 1 0 ± 40 0 ± 10 − 1            Corollary There is a 43-dimensional polytope with 86 facets and diameter (at least) 44. 19

  33. 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. 20

  34. 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

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

  36. 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: Products of several copies of it (dimension increases). 1 Gluing (or, rather, “blending”) several copies of it 2 (dimension is fixed). 22

  37. 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: Products of several copies of it (dimension increases). 1 Gluing (or, rather, “blending”) several copies of it 2 (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 n − d − 1 = δ − ( n − d ) δ ǫ ( P ) := . n − d 22

  38. 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: Products of several copies of it (dimension increases). 1 Gluing (or, rather, “blending”) several copies of it 2 (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 n − d − 1 = δ − ( n − d ) δ ǫ ( P ) := . n − d E. g.: The excess of our non-Hirsch polytope with n − d = 20 and with diameter 21 is 21 − 20 = 5 % . 20 22

  39. The Hirsch Conjecture Counter-examples to Hirsch Simplicial complexes Normal complexes How far can we go Many non-Hirsch polytopes Taking products preserves the excess. 1 Corollary For each k ∈ N there is a non-Hirsch polytope of dimension 20 k with 40 k facets and with excess 0 . 05 . 23

  40. The Hirsch Conjecture Counter-examples to Hirsch Simplicial complexes Normal complexes How far can we go Many non-Hirsch polytopes Taking products preserves the excess. 1 Corollary For each k ∈ N there is a non-Hirsch polytope of dimension 20 k with 40 k facets and with excess 0 . 05 . Gluing several copies (slightly) decreases the excess. 2 23

  41. The Hirsch Conjecture Counter-examples to Hirsch Simplicial complexes Normal complexes How far can we go Many non-Hirsch polytopes Taking products preserves the excess. 1 Corollary For each k ∈ N there is a non-Hirsch polytope of dimension 20 k with 40 k facets and with excess 0 . 05 . Gluing several copies (slightly) decreases the excess. 2 Corollary For each k ∈ N there is an infinite family of non-Hirsch polytopes of fixed dimension 20 k and with excess (tending to) � � 1 − 1 0 . 05 . k 23

  42. 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

  43. 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

  44. 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 � [ n ] � is a subset of . That is, a family of size d subsets of [ n ] := { 1 , . . . , n } . d The elements of K are called facets. Any subset of a facet is a face of K . 24

  45. 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 � [ n ] � is a subset of . That is, a family of size d subsets of [ n ] := { 1 , . . . , n } . d 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. 24

  46. The Hirsch Conjecture Counter-examples to Hirsch Simplicial complexes Normal complexes General simplicial complexes Several versions of the question: 25

  47. The Hirsch Conjecture Counter-examples to Hirsch Simplicial complexes Normal complexes General simplicial complexes Several versions of the question: Pure simplicial complexes, in general. H c ( d , n ) 25

  48. The Hirsch Conjecture Counter-examples to Hirsch Simplicial complexes Normal complexes General simplicial complexes Several versions of the question: Pure simplicial complexes, in general. H c ( d , n ) Pseudo-manifolds (w. or wo. bdry). H pm ( d , n ) , H pm ( d , n ) 25

  49. The Hirsch Conjecture Counter-examples to Hirsch Simplicial complexes Normal complexes General simplicial complexes Several versions of the question: Pure simplicial complexes, in general. H c ( d , n ) Pseudo-manifolds (w. or wo. bdry). H pm ( d , n ) , H pm ( d , n ) Manifolds (w. or wo. bdry). H M ( d , n ) , H M ( d , n ) 25

  50. The Hirsch Conjecture Counter-examples to Hirsch Simplicial complexes Normal complexes General simplicial complexes Several versions of the question: Pure simplicial complexes, in general. H c ( d , n ) Pseudo-manifolds (w. or wo. bdry). H pm ( d , n ) , H pm ( d , n ) Manifolds (w. or wo. bdry). H M ( d , n ) , H M ( d , n ) Spheres (or balls). H S ( d , n ) , H B ( d , n ) , . . . 25

  51. The Hirsch Conjecture Counter-examples to Hirsch Simplicial complexes Normal complexes General simplicial complexes Several versions of the question: Pure simplicial complexes, in general. H c ( d , n ) Pseudo-manifolds (w. or wo. bdry). H pm ( d , n ) , H pm ( d , n ) Manifolds (w. or wo. bdry). H M ( d , n ) , H M ( d , n ) Spheres (or balls). H S ( d , n ) , H B ( d , n ) , . . . H ∗ ( d , n ) is the maximum (dual) diameter; two simplices are considered adjacent if they differ by a single vertex. 25

  52. The Hirsch Conjecture Counter-examples to Hirsch Simplicial complexes Normal complexes General simplicial complexes Several versions of the question: Pure simplicial complexes, in general. H c ( d , n ) Pseudo-manifolds (w. or wo. bdry). H pm ( d , n ) , H pm ( d , n ) Manifolds (w. or wo. bdry). H M ( d , n ) , H M ( d , n ) Spheres (or balls). H S ( d , n ) , H B ( d , n ) , . . . H ∗ ( d , n ) is the maximum (dual) diameter; two simplices are considered adjacent if they differ by a single vertex. Lemma H c ( 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. 25

  53. 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 < H c ( 3 , n ) < 1 4 n 2 . 26

  54. 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 < H c ( 3 , n ) < 1 4 n 2 . In higher dimension: Theorem (S., 2013) H c ( md , mn ) > 2 2 m H c ( d , n ) m , ∀ m ∈ N . 26

  55. 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) 9 ( n − 1 ) 2 < H c ( 3 , n ) < 1 2 4 n 2 . In higher dimension: Theorem (S., 2013) H c ( md , mn ) > 2 2 m H c ( d , n ) m , ∀ m ∈ N . Corollary (S., 2013) Ω( n 2 d / 3 ) ≤ H c ( d , n ) ≤ O ( n d ) . 26

  56. The Hirsch Conjecture Counter-examples to Hirsch Simplicial complexes Normal complexes General simplicial complexes H c ( 3 , n ) > 2 9 ( n − 1 ) 2 Proof: Without loss of generality assume n = 3 k + 1. 1 27

  57. The Hirsch Conjecture Counter-examples to Hirsch Simplicial complexes Normal complexes General simplicial complexes H c ( 3 , n ) > 2 9 ( n − 1 ) 2 Proof: Without loss of generality assume n = 3 k + 1. 1 With the first 2 k + 1 vertices, construct k disjoint cycles of 2 length 2 k + 1 27

  58. The Hirsch Conjecture Counter-examples to Hirsch Simplicial complexes Normal complexes General simplicial complexes H c ( 3 , n ) > 2 9 ( n − 1 ) 2 Proof: Without loss of generality assume n = 3 k + 1. 1 With the first 2 k + 1 vertices, construct k disjoint cycles of 2 length 2 k + 1 (That is, decompose K 2 k + 1 into k disjoint Hamiltonian cycles). 27

  59. The Hirsch Conjecture Counter-examples to Hirsch Simplicial complexes Normal complexes General simplicial complexes H c ( 3 , n ) > 2 9 ( n − 1 ) 2 Proof: Without loss of generality assume n = 3 k + 1. 1 With the first 2 k + 1 vertices, construct k disjoint cycles of 2 length 2 k + 1 (That is, decompose K 2 k + 1 into k disjoint Hamiltonian cycles). Remove an edge from each cycle to make it a path, and 3 join each path to each of the remaining k vertices. 27

  60. The Hirsch Conjecture Counter-examples to Hirsch Simplicial complexes Normal complexes General simplicial complexes H c ( 3 , n ) > 2 9 ( n − 1 ) 2 Proof: Without loss of generality assume n = 3 k + 1. 1 With the first 2 k + 1 vertices, construct k disjoint cycles of 2 length 2 k + 1 (That is, decompose K 2 k + 1 into k disjoint Hamiltonian cycles). Remove an edge from each cycle to make it a path, and 3 join each path to each of the remaining k vertices. Glue together the k chains using k − 1 triangles. 4 27

  61. The Hirsch Conjecture Counter-examples to Hirsch Simplicial complexes Normal complexes General simplicial complexes H c ( 3 , n ) > 2 9 ( n − 1 ) 2 Proof: Without loss of generality assume n = 3 k + 1. 1 With the first 2 k + 1 vertices, construct k disjoint cycles of 2 length 2 k + 1 (That is, decompose K 2 k + 1 into k disjoint Hamiltonian cycles). Remove an edge from each cycle to make it a path, and 3 join each path to each of the remaining k vertices. Glue together the k chains using k − 1 triangles. 4 In this way we get a chain of triangles of length ( 2 k + 1 ) k − 2 > 2 9 ( 3 k ) 2 . 27

  62. The Hirsch Conjecture Counter-examples to Hirsch Simplicial complexes Normal complexes General simplicial complexes H c ( 3 , n ) > 2 9 ( n − 1 ) 2 28

  63. The Hirsch Conjecture Counter-examples to Hirsch Simplicial complexes Normal complexes General simplicial complexes H c ( md , mn ) > 1 2 H c ( d , n ) m Proof: Let K be a complex achieving H C ( d , n ) . W.l.o.g. assume 1 its dual graph is a path. 29

  64. The Hirsch Conjecture Counter-examples to Hirsch Simplicial complexes Normal complexes General simplicial complexes H c ( md , mn ) > 1 2 H c ( d , n ) m Proof: Let K be a complex achieving H C ( d , n ) . W.l.o.g. assume 1 its dual graph is a path. Take the join K ∗ m := K ∗ K ∗ · · · ∗ K of m copies of K . 2 29

  65. The Hirsch Conjecture Counter-examples to Hirsch Simplicial complexes Normal complexes General simplicial complexes H c ( md , mn ) > 1 2 H c ( d , n ) m Proof: Let K be a complex achieving H C ( d , n ) . W.l.o.g. assume 1 its dual graph is a path. Take the join K ∗ m := K ∗ K ∗ · · · ∗ K of m copies of K . 2 K ∗ m is a complex of dimension md − 1, with mn vertices and whose dual graph is an m -dimensional grid of side H C ( d , n ) . (It has ( H C ( d , n ) + 1 ) m facets). 29

  66. The Hirsch Conjecture Counter-examples to Hirsch Simplicial complexes Normal complexes General simplicial complexes H c ( md , mn ) > 1 2 H c ( d , n ) m Proof: Let K be a complex achieving H C ( d , n ) . W.l.o.g. assume 1 its dual graph is a path. Take the join K ∗ m := K ∗ K ∗ · · · ∗ K of m copies of K . 2 K ∗ m is a complex of dimension md − 1, with mn vertices and whose dual graph is an m -dimensional grid of side H C ( d , n ) . (It has ( H C ( d , n ) + 1 ) m facets). In this grid consider a maximal induced path. This can be 3 done using more than half of the vertices. 29

  67. The Hirsch Conjecture Counter-examples to Hirsch Simplicial complexes Normal complexes General simplicial complexes H c ( md , mn ) > 1 2 H c ( d , n ) m 30

  68. 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? 31

  69. 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. 31

  70. 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 only facets that contain X ∩ Y . 31

  71. 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 only facets that contain X ∩ Y . Adler and Dantzig (1974) call a normal pseudo-manifold an abstract polytope 31

  72. 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

  73. 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

  74. 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

  75. 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

  76. 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 H n ( d , n ) ≤ n log d + 1 , H n ( d , n ) ≤ 2 d − 2 n . 32

  77. 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

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