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arXiv · 2611.08819

Reflexive polytopes can have doubly exponentially many vertices

Johannes Hofscheier·Benjamin Nill
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Abstract

Reflexive polytopes are lattice polytopes containing the origin in their interior such that also the dual polytope is a lattice polytope. On the basis of low-dimensional observations and of bounds known for special classes of reflexive polytopes, the second author conjectured in 2004 that the number of vertices of $d$-dimensional reflexive polytopes is bounded from above by $6^{d/2}$. In this note, we present counterexamples showing that this conjecture is false in every dimension $d \ge 8$. In fact, rather surprisingly, in sufficiently large dimensions $d$ the number of vertices (equivalently, by duality, the number of facets) can be as large as theoretically possible, namely doubly exponential in $d$. These results were found using GPT-6.1 Sol.

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Johannes Hofscheier, Benjamin Nill. 2026-11-06. Reflexive polytopes can have doubly exponentially many vertices. https://arxiv.org/abs/2611.08819

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