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The counter-example(s) Asymptotic diameter Simplicial complexes Connected layer families How false is the Hirsch Conjecture? Francisco Santos http://personales.unican.es/santosf Departamento de Matemticas, Estadstica y Computacin

  1. The counter-example(s) Asymptotic diameter Simplicial complexes Connected layer families 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, 2011]. 5-prismatoids of width 6 exist [S., 2010] with 25 vertices [Matschke-S.-Weibel 2011+]. 5-prismatoids of arbitrarily large width exist [Matschke-S.-Weibel 2011+]. 9

  2. The counter-example(s) Asymptotic diameter Simplicial complexes Connected layer families 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, 2011]. 5-prismatoids of width 6 exist [S., 2010] with 25 vertices [Matschke-S.-Weibel 2011+]. 5-prismatoids of arbitrarily large width exist [Matschke-S.-Weibel 2011+]. 9

  3. The counter-example(s) Asymptotic diameter Simplicial complexes Connected layer families 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, 2011]. 5-prismatoids of width 6 exist [S., 2010] with 25 vertices [Matschke-S.-Weibel 2011+]. 5-prismatoids of arbitrarily large width exist [Matschke-S.-Weibel 2011+]. 9

  4. The counter-example(s) Asymptotic diameter Simplicial complexes Connected layer families 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, 2011]. 5-prismatoids of width 6 exist [S., 2010] with 25 vertices [Matschke-S.-Weibel 2011+]. 5-prismatoids of arbitrarily large width exist [Matschke-S.-Weibel 2011+]. 9

  5. The counter-example(s) Asymptotic diameter Simplicial complexes Connected layer families 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, 2011]. 5-prismatoids of width 6 exist [S., 2010] with 25 vertices [Matschke-S.-Weibel 2011+]. 5-prismatoids of arbitrarily large width exist [Matschke-S.-Weibel 2011+]. 9

  6. The counter-example(s) Asymptotic diameter Simplicial complexes Connected layer families A 5-prismatoid of width > 5 Theorem The following prismatoid Q, of dimension 5 and with 48 vertices, has width six. 10

  7. The counter-example(s) Asymptotic diameter Simplicial complexes Connected layer families A 5-prismatoid of width > 5 Theorem 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 8 9 > > > > ± 18 0 0 0 1 0 0 0 ± 18 − 1 > 0 1 0 1 > > > > > > > 0 ± 18 0 0 1 0 0 ± 18 0 − 1 > > > > > B C B C > > > 0 0 ± 45 0 1 ± 45 0 0 0 − 1 > B C B C > > > > B C B C > > > 0 0 0 ± 45 1 0 ± 45 0 0 − 1 < B C B C = Q := conv B C B C ± 15 ± 15 0 0 1 0 0 ± 15 ± 15 − 1 B C B C > B C B C > > 0 0 ± 30 ± 30 1 ± 30 ± 30 0 0 − 1 > > B C B C > > > > B C B C > > 0 ± 10 ± 40 0 1 ± 40 0 ± 10 0 − 1 > > @ A @ A > > > > > > ± 10 0 0 ± 40 1 0 ± 40 0 ± 10 − 1 > > > > > > > > > : ; 10

  8. The counter-example(s) Asymptotic diameter Simplicial complexes Connected layer families A 5-prismatoid of width > 5 Theorem The following prismatoid Q, of dimension 5 and with 48 vertices, has width six. Corollary There is a 43-dimensional polytope with 86 facets and diameter (at least) 44. 10

  9. The counter-example(s) Asymptotic diameter Simplicial complexes Connected layer families Combinatorics of prismatoids Proof. Analyzing the combinatorics of a d -prismatoid Q can be done via an intermediate slice . . . Q + Q Q ∩ H H Q − 11

  10. The counter-example(s) Asymptotic diameter Simplicial complexes Connected layer families Combinatorics of prismatoids Proof. . . . which equals the Minkowski sum Q + + Q − of the two bases Q + and Q − . The normal fan of Q + + Q − equals the “superposi- tion” of those of Q + and Q − . 1 + 1 = 2 2 11

  11. The counter-example(s) Asymptotic diameter Simplicial complexes Connected layer families Combinatorics of prismatoids Proof. . . . which equals the Minkowski sum Q + + Q − of the two bases Q + and Q − . The normal fan of Q + + Q − equals the “superposi- tion” of those of Q + and Q − . 1 + 1 = 2 2 11

  12. The counter-example(s) Asymptotic diameter Simplicial complexes Connected layer families 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 (“map”) of the d − 2-sphere. Theorem Let Q be a d-prismatoid with bases Q + and Q − and let G + and G − be the corresponding maps in the ( d − 2 ) -sphere ( central projection of the normal fans of Q + and Q − ). Then, the width of Q equals 2 plus the minimum number of steps needed to go from a vertex of G + to a vertex of G − in the (graph of) the superposition of the two maps. 12

  13. The counter-example(s) Asymptotic diameter Simplicial complexes Connected layer families 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 (“map”) of the d − 2-sphere. Theorem Let Q be a d-prismatoid with bases Q + and Q − and let G + and G − be the corresponding maps in the ( d − 2 ) -sphere ( central projection of the normal fans of Q + and Q − ). Then, the width of Q equals 2 plus the minimum number of steps needed to go from a vertex of G + to a vertex of G − in the (graph of) the superposition of the two maps. 12

  14. The counter-example(s) Asymptotic diameter Simplicial complexes Connected layer families 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 (“map”) of the d − 2-sphere. Theorem Let Q be a d-prismatoid with bases Q + and Q − and let G + and G − be the corresponding maps in the ( d − 2 ) -sphere ( central projection of the normal fans of Q + and Q − ). Then, the width of Q equals 2 plus the minimum number of steps needed to go from a vertex of G + to a vertex of G − in the (graph of) the superposition of the two maps. 12

  15. The counter-example(s) Asymptotic diameter Simplicial complexes Connected layer families 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 (“map”) of the d − 2-sphere. Theorem Let Q be a d-prismatoid with bases Q + and Q − and let G + and G − be the corresponding maps in the ( d − 2 ) -sphere ( central projection of the normal fans of Q + and Q − ). Then, the width of Q equals 2 plus the minimum number of steps needed to go from a vertex of G + to a vertex of G − in the (graph of) the superposition of the two maps. 12

  16. The counter-example(s) Asymptotic diameter Simplicial complexes Connected layer families A 4-prismatoid of width > 4? Replicating the following basic pattern we obtain a periodic pair of maps in the plain that is “non-Hirsch”. 13

  17. The counter-example(s) Asymptotic diameter Simplicial complexes Connected layer families A 4-prismatoid of width > 4? Replicating the following basic pattern we obtain a periodic pair of maps in the plain that is “non-Hirsch”. 13

  18. The counter-example(s) Asymptotic diameter Simplicial complexes Connected layer families A 4-prismatoid of width > 4? Replicating the following basic pattern we obtain a periodic pair of maps in the plain that is “non-Hirsch”. 13

  19. The counter-example(s) Asymptotic diameter Simplicial complexes Connected layer families A 4-prismatoid of width > 4? Replicating the following basic pattern we obtain a periodic pair of maps in the plain that is “non-Hirsch”. 13

  20. The counter-example(s) Asymptotic diameter Simplicial complexes Connected layer families A 4-prismatoid of width > 4? Replicating the following basic pattern we obtain a periodic pair of maps in the plain that is “non-Hirsch”. 13

  21. The counter-example(s) Asymptotic diameter Simplicial complexes Connected layer families A 4-prismatoid of width > 4? Replicating the following basic pattern we obtain a periodic pair of maps in the plain that is “non-Hirsch”. 13

  22. The counter-example(s) Asymptotic diameter Simplicial complexes Connected layer families A 4-prismatoid of width > 4? Replicating the following basic pattern we obtain a periodic pair of maps in the plain that is “non-Hirsch”. If this drawing was on a 2-sphere it would represent a 4- prismatoid of width 5. 13

  23. The counter-example(s) Asymptotic diameter Simplicial complexes Connected layer families A 4-prismatoid of width > 4? Replicating the following basic pattern we obtain a periodic pair of maps in the plain that is “non-Hirsch”. If this drawing was on a 2-sphere it would represent a 4- prismatoid of width 5. This does not work, but putting the drawing in (two tori embed- ded in) S 3 does, and gives a prismatoid with 48 vertices. 13

  24. The counter-example(s) Asymptotic diameter Simplicial complexes Connected layer families A 5-prismatoid of width > 5 14

  25. The counter-example(s) Asymptotic diameter Simplicial complexes Connected layer families A 5-prismatoid of width > 5 14

  26. The counter-example(s) Asymptotic diameter Simplicial complexes Connected layer families Smaller 5-prismatoids of width > 5 With the same ideas Theorem (Matschke-Santos-Weibel, 2011) 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. 15

  27. The counter-example(s) Asymptotic diameter Simplicial complexes Connected layer families Smaller 5-prismatoids of width > 5 With the same ideas Theorem (Matschke-Santos-Weibel, 2011) 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. 15

  28. The counter-example(s) Asymptotic diameter Simplicial complexes Connected layer families Smaller 5-prismatoids of width > 5 With the same ideas Theorem (Matschke-Santos-Weibel, 2011) 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. 15

  29. The counter-example(s) Asymptotic diameter Simplicial complexes Connected layer families 16

  30. The counter-example(s) Asymptotic diameter Simplicial complexes Connected layer families Asymptotic width in dimension five Theorem There are 5 -dimensional prismatoids with n vertices and width Ω( √ n ) . Sketch of proof Apply the same technique, with this other pair of maps. To embed it in S 3 you need quadratically many tetrahedra. 17

  31. The counter-example(s) Asymptotic diameter Simplicial complexes Connected layer families Asymptotic width in dimension five Theorem There are 5 -dimensional prismatoids with n vertices and width Ω( √ n ) . Sketch of proof Apply the same technique, with this other pair of maps. To embed it in S 3 you need quadratically many tetrahedra. 17

  32. The counter-example(s) Asymptotic diameter Simplicial complexes Connected layer families Asymptotic width in dimension five Theorem There are 5 -dimensional prismatoids with n vertices and width Ω( √ n ) . Sketch of proof Apply the same technique, with this other pair of maps. To embed it in S 3 you need quadratically many tetrahedra. 17

  33. The counter-example(s) Asymptotic diameter Simplicial complexes Connected layer families Asymptotic width in dimension five Theorem There are 5 -dimensional prismatoids with n vertices and width Ω( √ n ) . Sketch of proof Apply the same technique, with this other pair of maps. To embed it in S 3 you need quadratically many tetrahedra. 17

  34. The counter-example(s) Asymptotic diameter Simplicial complexes Connected layer families Asymptotic width in dimension five Theorem There are 5 -dimensional prismatoids with n vertices and width Ω( √ n ) . Sketch of proof Apply the same technique, with this other pair of maps. To embed it in S 3 you need quadratically many tetrahedra. 17

  35. The counter-example(s) Asymptotic diameter Simplicial complexes Connected layer families Asymptotic width in dimension five Theorem There are 5 -dimensional prismatoids with n vertices and width Ω( √ n ) . Sketch of proof Apply the same technique, with this other pair of maps. To embed it in S 3 you need quadratically many tetrahedra. 17

  36. The counter-example(s) Asymptotic diameter Simplicial complexes Connected layer families 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 several copies of it (dimension is fixed). 2 To analyze the asymptotics of these operations, we call excess of a d -polytope with n facets and diameter δ the number n − d − 1 = δ − ( n − d ) δ . n − d E. g. , the excess of the non-Hirsch polytope of dimension 20 with diameter 21 is 21 − 20 = 5 % . 20 18

  37. The counter-example(s) Asymptotic diameter Simplicial complexes Connected layer families 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 several copies of it (dimension is fixed). 2 To analyze the asymptotics of these operations, we call excess of a d -polytope with n facets and diameter δ the number n − d − 1 = δ − ( n − d ) δ . n − d E. g. , the excess of the non-Hirsch polytope of dimension 20 with diameter 21 is 21 − 20 = 5 % . 20 18

  38. The counter-example(s) Asymptotic diameter Simplicial complexes Connected layer families 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 several copies of it (dimension is fixed). 2 To analyze the asymptotics of these operations, we call excess of a d -polytope with n facets and diameter δ the number n − d − 1 = δ − ( n − d ) δ . n − d E. g. , the excess of the non-Hirsch polytope of dimension 20 with diameter 21 is 21 − 20 = 5 % . 20 18

  39. The counter-example(s) Asymptotic diameter Simplicial complexes Connected layer families 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 several copies of it (dimension is fixed). 2 To analyze the asymptotics of these operations, we call excess of a d -polytope with n facets and diameter δ the number n − d − 1 = δ − ( n − d ) δ . n − d E. g. , the excess of the non-Hirsch polytope of dimension 20 with diameter 21 is 21 − 20 = 5 % . 20 18

  40. The counter-example(s) Asymptotic diameter Simplicial complexes Connected layer families 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 several copies of it (dimension is fixed). 2 To analyze the asymptotics of these operations, we call excess of a d -polytope with n facets and diameter δ the number n − d − 1 = δ − ( n − d ) δ . n − d E. g. , the excess of the non-Hirsch polytope of dimension 20 with diameter 21 is 21 − 20 = 5 % . 20 18

  41. The counter-example(s) Asymptotic diameter Simplicial complexes Connected layer families 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 several copies of it (dimension is fixed). 2 To analyze the asymptotics of these operations, we call excess of a d -polytope with n facets and diameter δ the number n − d − 1 = δ − ( n − d ) δ . n − d E. g. , the excess of the non-Hirsch polytope of dimension 20 with diameter 21 is 21 − 20 = 5 % . 20 18

  42. The counter-example(s) Asymptotic diameter Simplicial complexes Connected layer families Many non-Hirsch polytopes Taking products preserves the excess: for each k ∈ N , 1 there is a non-Hirsch polytope of dimension 20 k with 40 k facets and with excess equal to 0 . 05 = 5 % . Gluing several copies (slightly) decreases the excess. 2 19

  43. The counter-example(s) Asymptotic diameter Simplicial complexes Connected layer families Many non-Hirsch polytopes Taking products preserves the excess: for each k ∈ N , 1 there is a non-Hirsch polytope of dimension 20 k with 40 k facets and with excess equal to 0 . 05 = 5 % . Gluing several copies (slightly) decreases the excess. 2 19

  44. The counter-example(s) Asymptotic diameter Simplicial complexes Connected layer families Many non-Hirsch polytopes Taking products preserves the excess: for each k ∈ N , 1 there is a non-Hirsch polytope of dimension 20 k with 40 k facets and with excess equal to 0 . 05 = 5 % . Gluing several copies (slightly) decreases the excess. 2 19

  45. The counter-example(s) Asymptotic diameter Simplicial complexes Connected layer families Many non-Hirsch polytopes Taking products preserves the excess: for each k ∈ N , 1 there is a non-Hirsch polytope of dimension 20 k with 40 k facets and with excess equal to 0 . 05 = 5 % . Gluing several copies (slightly) decreases the excess. 2 n − d = ( n 1 + n 2 − d ) − d = ( n 1 − d ) + ( n 2 − d ) δ = δ 1 + δ 2 − 1 19

  46. The counter-example(s) Asymptotic diameter Simplicial complexes Connected layer families Many non-Hirsch polytopes Taking products preserves the excess: for each k ∈ N , 1 there is a non-Hirsch polytope of dimension 20 k with 40 k facets and with excess equal to 0 . 05 = 5 % . 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 (more or less) � � 1 − 1 0 . 05 . k 19

  47. The counter-example(s) Asymptotic diameter Simplicial complexes Connected layer families Prismatoids of large width won’t help (much) But we know there are “worse” prismatoids: 5-prismatoids of arbitrarily large width. Will those produce non-Hirsch polytopes with more excess? Let us be optimistic and suppose that we could construct 5-prismatoids with n vertices and linear width ≃ α n (the examples we have have width Θ( √ n ) ) . The non-Hirsch polytopes derived from them would have: Dimension D = n − d ≃ n . Number of facets N = 2 n − 2 d ≃ 2 n Diameter δ = α n + ( n − 2 d ) ≃ ( 1 + α ) n Their asymptotic excess is: n − d − 1 ≃ ( 1 + α ) n δ − 1 = α. n 20

  48. The counter-example(s) Asymptotic diameter Simplicial complexes Connected layer families Prismatoids of large width won’t help (much) But we know there are “worse” prismatoids: 5-prismatoids of arbitrarily large width. Will those produce non-Hirsch polytopes with more excess? Let us be optimistic and suppose that we could construct 5-prismatoids with n vertices and linear width ≃ α n (the examples we have have width Θ( √ n ) ) . The non-Hirsch polytopes derived from them would have: Dimension D = n − d ≃ n . Number of facets N = 2 n − 2 d ≃ 2 n Diameter δ = α n + ( n − 2 d ) ≃ ( 1 + α ) n Their asymptotic excess is: n − d − 1 ≃ ( 1 + α ) n δ − 1 = α. n 20

  49. The counter-example(s) Asymptotic diameter Simplicial complexes Connected layer families Prismatoids of large width won’t help (much) But we know there are “worse” prismatoids: 5-prismatoids of arbitrarily large width. Will those produce non-Hirsch polytopes with more excess? Let us be optimistic and suppose that we could construct 5-prismatoids with n vertices and linear width ≃ α n (the examples we have have width Θ( √ n ) ) . The non-Hirsch polytopes derived from them would have: Dimension D = n − d ≃ n . Number of facets N = 2 n − 2 d ≃ 2 n Diameter δ = α n + ( n − 2 d ) ≃ ( 1 + α ) n Their asymptotic excess is: n − d − 1 ≃ ( 1 + α ) n δ − 1 = α. n 20

  50. The counter-example(s) Asymptotic diameter Simplicial complexes Connected layer families Prismatoids of large width won’t help (much) But we know there are “worse” prismatoids: 5-prismatoids of arbitrarily large width. Will those produce non-Hirsch polytopes with more excess? Let us be optimistic and suppose that we could construct 5-prismatoids with n vertices and linear width ≃ α n (the examples we have have width Θ( √ n ) ) . The non-Hirsch polytopes derived from them would have: Dimension D = n − d ≃ n . Number of facets N = 2 n − 2 d ≃ 2 n Diameter δ = α n + ( n − 2 d ) ≃ ( 1 + α ) n Their asymptotic excess is: n − d − 1 ≃ ( 1 + α ) n δ − 1 = α. n 20

  51. The counter-example(s) Asymptotic diameter Simplicial complexes Connected layer families Revenge of the linear bound OK, can we hope for prismatoids of width greater than linear? In fixed dimension, certainly not: Theorem The width of a d-dimensional prismatoid with n vertices cannot exceed 2 d − 3 n. Proof. This is a general result for the (dual) diameter of a polytope [Barnette, Larman, ∼ 1970]. 21

  52. The counter-example(s) Asymptotic diameter Simplicial complexes Connected layer families Revenge of the linear bound OK, can we hope for prismatoids of width greater than linear? In fixed dimension, certainly not: Theorem The width of a d-dimensional prismatoid with n vertices cannot exceed 2 d − 3 n. Proof. This is a general result for the (dual) diameter of a polytope [Barnette, Larman, ∼ 1970]. 21

  53. The counter-example(s) Asymptotic diameter Simplicial complexes Connected layer families Revenge of the linear bound OK, can we hope for prismatoids of width greater than linear? In fixed dimension, certainly not: Theorem The width of a d-dimensional prismatoid with n vertices cannot exceed 2 d − 3 n. Proof. This is a general result for the (dual) diameter of a polytope [Barnette, Larman, ∼ 1970]. 21

  54. The counter-example(s) Asymptotic diameter Simplicial complexes Connected layer families Revenge of the linear bound OK, can we hope for prismatoids of width greater than linear? In fixed dimension, certainly not: Theorem The width of a d-dimensional prismatoid with n vertices cannot exceed 2 d − 3 n. Proof. This is a general result for the (dual) diameter of a polytope [Barnette, Larman, ∼ 1970]. 21

  55. The counter-example(s) Asymptotic diameter Simplicial complexes Connected layer families Revenge of the linear bound In fact, in dimension five we can tighten the upper bound a little bit: Theorem The width of a 5 -dimensional prismatoid with n vertices cannot exceed n / 2 + 3 . 22

  56. The counter-example(s) Asymptotic diameter Simplicial complexes Connected layer families Revenge of the linear bound In fact, in dimension five we can tighten the upper bound a little bit: Theorem The width of a 5 -dimensional prismatoid with n vertices cannot exceed n / 2 + 3 . 22

  57. The counter-example(s) Asymptotic diameter Simplicial complexes Connected layer families Revenge of the linear bound In fact, in dimension five we can tighten the upper bound a little bit: Theorem The width of a 5 -dimensional prismatoid with n vertices cannot exceed n / 2 + 3 . 22

  58. The counter-example(s) Asymptotic diameter Simplicial complexes Connected layer families Revenge of the linear bound In fact, in dimension five we can tighten the upper bound a little bit: Theorem The width of a 5 -dimensional prismatoid with n vertices cannot exceed n / 2 + 3 . 22

  59. The counter-example(s) Asymptotic diameter Simplicial complexes Connected layer families Revenge of the linear bound In fact, in dimension five we can tighten the upper bound a little bit: Theorem The width of a 5 -dimensional prismatoid with n vertices cannot exceed n / 2 + 3 . 22

  60. The counter-example(s) Asymptotic diameter Simplicial complexes Connected layer families Revenge of the linear bound In fact, in dimension five we can tighten the upper bound a little bit: Theorem The width of a 5 -dimensional prismatoid with n vertices cannot exceed n / 2 + 3 . Corollary Using the Strong d-step Theorem for 5-prismatoids it is impossible to violate the Hirsch conjecture by more than 50 % . 22

  61. The counter-example(s) Asymptotic diameter Simplicial complexes Connected layer families More general setting Instead of looking at (simplicial) polytopes, why not look at the maximum diameter of more general complexes? Strongly connected pure simplicial complexes. H C ( n , d ) Pseudo-manifolds (w. or wo. bdry). H pm ( n , d ) , H pm ( n , d ) Simplicial manifolds (w. or wo. bdry). H M ( n , d ) , H M ( n , d )

  62. The counter-example(s) Asymptotic diameter Simplicial complexes Connected layer families More general setting Instead of looking at (simplicial) polytopes, why not look at the maximum diameter of more general complexes? Strongly connected pure simplicial complexes. H C ( n , d ) Pseudo-manifolds (w. or wo. bdry). H pm ( n , d ) , H pm ( n , d ) Simplicial manifolds (w. or wo. bdry). H M ( n , d ) , H M ( n , d ) Simplicial spheres (or balls). H S ( n , d ) , H B ( n , d ) , . . . Remark, in all definitions of H • ( n , d ) , n is the number of vertices and d − 1 is the dimension. 23

  63. The counter-example(s) Asymptotic diameter Simplicial complexes Connected layer families More general setting Instead of looking at (simplicial) polytopes, why not look at the maximum diameter of more general complexes? Strongly connected pure simplicial complexes. H C ( n , d ) Pseudo-manifolds (w. or wo. bdry). H pm ( n , d ) , H pm ( n , d ) Simplicial manifolds (w. or wo. bdry). H M ( n , d ) , H M ( n , d ) Simplicial spheres (or balls). H S ( n , d ) , H B ( n , d ) , . . . Remark, in all definitions of H • ( n , d ) , n is the number of vertices and d − 1 is the dimension. 23

  64. The counter-example(s) Asymptotic diameter Simplicial complexes Connected layer families More general setting Instead of looking at (simplicial) polytopes, why not look at the maximum diameter of more general complexes? Strongly connected pure simplicial complexes. H C ( n , d ) Pseudo-manifolds (w. or wo. bdry). H pm ( n , d ) , H pm ( n , d ) Simplicial manifolds (w. or wo. bdry). H M ( n , d ) , H M ( n , d ) Simplicial spheres (or balls). H S ( n , d ) , H B ( n , d ) , . . . Remark, in all definitions of H • ( n , d ) , n is the number of vertices and d − 1 is the dimension. 23

  65. The counter-example(s) Asymptotic diameter Simplicial complexes Connected layer families More general setting Instead of looking at (simplicial) polytopes, why not look at the maximum diameter of more general complexes? Strongly connected pure simplicial complexes. H C ( n , d ) Pseudo-manifolds (w. or wo. bdry). H pm ( n , d ) , H pm ( n , d ) Simplicial manifolds (w. or wo. bdry). H M ( n , d ) , H M ( n , d ) Simplicial spheres (or balls). H S ( n , d ) , H B ( n , d ) , . . . Remark, in all definitions of H • ( n , d ) , n is the number of vertices and d − 1 is the dimension. 23

  66. The counter-example(s) Asymptotic diameter Simplicial complexes Connected layer families More general setting Instead of looking at (simplicial) polytopes, why not look at the maximum diameter of more general complexes? Strongly connected pure simplicial complexes. H C ( n , d ) Pseudo-manifolds (w. or wo. bdry). H pm ( n , d ) , H pm ( n , d ) Simplicial manifolds (w. or wo. bdry). H M ( n , d ) , H M ( n , d ) Simplicial spheres (or balls). H S ( n , d ) , H B ( n , d ) , . . . Remark, in all definitions of H • ( n , d ) , n is the number of vertices and d − 1 is the dimension. 23

  67. The counter-example(s) Asymptotic diameter Simplicial complexes Connected layer families Some easy remarks and a toy example There are the following relations: H C ( n , d ) = H pm ( n , d ) ≥ H M ( n , d ) ≥ H B ( n , d ) VI VI VI H pm ( n , d ) ≥ H M ( n , d ) ≥ H S ( n , d ) In dimension one (graphs): H C ( n , 2 ) = H pm ( n , 2 ) = H M ( n , 2 ) = H B ( n , 2 ) = n − 1 , � n � H pm ( n , 2 ) = H M ( n , 2 ) = H S ( n , 2 ) = , 2 24

  68. The counter-example(s) Asymptotic diameter Simplicial complexes Connected layer families Some easy remarks and a toy example There are the following relations: H C ( n , d ) = H pm ( n , d ) ≥ H M ( n , d ) ≥ H B ( n , d ) VI VI VI H pm ( n , d ) ≥ H M ( n , d ) ≥ H S ( n , d ) In dimension one (graphs): H C ( n , 2 ) = H pm ( n , 2 ) = H M ( n , 2 ) = H B ( n , 2 ) = n − 1 , � n � H pm ( n , 2 ) = H M ( n , 2 ) = H S ( n , 2 ) = , 2 24

  69. The counter-example(s) Asymptotic diameter Simplicial complexes Connected layer families The maximum diameter of pure simplicial complexes In dimension two: Theorem 2 9 ( n − 1 ) 2 < H C ( n , 3 ) = H pm ( n , 3 ) < 1 4 n 2 . In higher dimension: Theorem H C ( kn , kd ) > 1 2 k H C ( n , d ) k . Corollary � � � � 2 d n n 3 Ω < H C ( n , d ) = H pm ( n , d ) < . d d − 1 9 3 25

  70. The counter-example(s) Asymptotic diameter Simplicial complexes Connected layer families The maximum diameter of pure simplicial complexes In dimension two: Theorem 2 9 ( n − 1 ) 2 < H C ( n , 3 ) = H pm ( n , 3 ) < 1 4 n 2 . In higher dimension: Theorem H C ( kn , kd ) > 1 2 k H C ( n , d ) k . Corollary � � � � 2 d n n 3 Ω < H C ( n , d ) = H pm ( n , d ) < . d d − 1 9 3 25

  71. The counter-example(s) Asymptotic diameter Simplicial complexes Connected layer families The maximum diameter of pure simplicial complexes In dimension two: Theorem 2 9 ( n − 1 ) 2 < H C ( n , 3 ) = H pm ( n , 3 ) < 1 4 n 2 . In higher dimension: Theorem H C ( kn , kd ) > 1 2 k H C ( n , d ) k . Corollary � � � � 2 d n n 3 Ω < H C ( n , d ) = H pm ( n , d ) < . d d − 1 9 3 25

  72. The counter-example(s) Asymptotic diameter Simplicial complexes Connected layer families The maximum diameter of pure simplicial complexes In dimension two: Theorem 2 9 ( n − 1 ) 2 < H C ( n , 3 ) = H pm ( n , 3 ) < 1 4 n 2 . In higher dimension: Theorem H C ( kn , kd ) > 1 2 k H C ( n , d ) k . Corollary � � � � 2 d n n 3 Ω < H C ( n , d ) = H pm ( n , d ) < . d d − 1 9 3 25

  73. The counter-example(s) Asymptotic diameter Simplicial complexes Connected layer families H pm ( n , 3 ) > 2 9 ( n − 1 ) 2 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 chain, and 3 join each chain 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 . 26

  74. The counter-example(s) Asymptotic diameter Simplicial complexes Connected layer families H pm ( n , 3 ) > 2 9 ( n − 1 ) 2 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 chain, and 3 join each chain 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 . 26

  75. The counter-example(s) Asymptotic diameter Simplicial complexes Connected layer families H pm ( n , 3 ) > 2 9 ( n − 1 ) 2 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 chain, and 3 join each chain 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 . 26

  76. The counter-example(s) Asymptotic diameter Simplicial complexes Connected layer families H pm ( n , 3 ) > 2 9 ( n − 1 ) 2 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 chain, and 3 join each chain 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 . 26

  77. The counter-example(s) Asymptotic diameter Simplicial complexes Connected layer families H pm ( n , 3 ) > 2 9 ( n − 1 ) 2 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 chain, and 3 join each chain 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 . 26

  78. The counter-example(s) Asymptotic diameter Simplicial complexes Connected layer families H pm ( n , 3 ) > 2 9 ( n − 1 ) 2 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 chain, and 3 join each chain 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 . 26

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