Counterexample to a Variant of a Conjecture of Ziegler Concerning a Simple Polytope and Its Dual

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 \(d \ge 3\).

1. https://polymake.org/polytopes/paffenholz/www/other.html.

The author thanks Richard Ehrenborg and Margaret Readdy for inspiring conversations and comments on an earlier draft, and Gábor Hetyei for comments. The author also thanks the referees for helpful remarks.

Authors and Affiliations

1. Department of Mathematics, University of Kentucky, Lexington, KY, 40506, USA

   William Gustafson

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Correspondence to William Gustafson.

Editor in Charge: János Pach

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