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

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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 $$d \ge 3$$ .