Counterexample to a Variant of a Conjecture of Ziegler Concerning a Simple Polytope and Its Dual
- PDF / 205,586 Bytes
- 3 Pages / 439.37 x 666.142 pts Page_size
- 113 Downloads / 197 Views
Counterexample to a Variant of a Conjecture of Ziegler Concerning a Simple Polytope and Its Dual William Gustafson1 Received: 16 January 2020 / Revised: 23 August 2020 / Accepted: 24 September 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract Problem 4.19 in Ziegler (Lectures on Polytopes. Graduate Texts in Mathematics, vol. 152. Springer, New York (1995)) asserts that every simple 3-dimensional polytope has the property that its dual can be constructed as the convex hull of points chosen from the facets of the original polytope. In this note we state a variant of this conjecture that requires the points to be a subset of the vertices of the original polytope, and provide a family of counterexamples for dimension d ≥ 3. Keywords Convex polytope · Dual polytope · Simple polytope Mathematics Subject Classification 52B05 · 52B11 · 05E99
1 Extended Conjecture and Counterexamples We begin with the following conjecture. We then provide an infinite number of counterexamples. Conjecture 1.1 Let P be a simple polytope of dimension greater than or equal to 3. Then there exists a subset S of the vertices of P such that the convex hull of S has the same combinatorial type as the dual polytope P ∗ . Ziegler’s original conjecture was to select points from the facets of a 3-dimensional simple polytope so that the convex hull has the same combinatorial type as the dual; see [1, Exercise 4.19, pp. 123–124]. Paffenholz1 verified Ziegler’s conjecture for the truncated tetrahedron, the truncated cube, and the truncated cross-polytope by giving an explicit realization. 1 https://polymake.org/polytopes/paffenholz/www/other.html.
Editor in Charge: János Pach William Gustafson [email protected] 1
Department of Mathematics, University of Kentucky, Lexington, KY 40506, USA
123
Discrete & Computational Geometry
Note that Conjecture 1.1 is true in dimensions at most 2 and is immediately true for any d-dimensional simplex. The conjecture holds for any d-dimensional cube, [0, 1]d as well. The convex hull of the set consisting of vertices (0, . . . , 0), (1, . . . , 1) and all vertices of the forms (0, . . . , 0, 1, . . . , 1) or (1, . . . , 1, 0, . . . , 0), forms a ddimensional cross-polytope. We will show that Conjecture 1.1 is false in all dimensions greater than or equal to 3. Theorem 1.2 Let d be a positive integer. Let P be a d -dimensional polytope with d ≥ 2 and n facets such that every vertex is incident with at most (n + 2d)/2d − d facets. If Q is the Cartesian product of P with the d-dimensional cube, then there is no subset S of the vertices of Q such that the convex hull of S is combinatorially equivalent to the dual polytope Q ∗ . Proof Suppose on the contrary that S is a subset of the vertices of the polytope Q satisfying that the convex hull of S is combinatorially equivalent to the dual polytope Q ∗ . Observe that the dual polytope Q ∗ is combinatorially equivalent to the d times iterated bipyramid over P ∗ and thus has n + 2d vertices. The vertices of S can then be divi
Data Loading...
