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

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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 \(d \ge 3\).

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Notes

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

References

  1. Ziegler, G.M.: Lectures on Polytopes. Graduate Texts in Mathematics, vol. 152. Springer, New York (1995)

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Acknowledgements

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.

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Correspondence to William Gustafson.

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Gustafson, W. Counterexample to a Variant of a Conjecture of Ziegler Concerning a Simple Polytope and Its Dual. Discrete Comput Geom 66, 1470–1472 (2021). https://doi.org/10.1007/s00454-020-00253-5

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  • DOI: https://doi.org/10.1007/s00454-020-00253-5

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