Reflexive Polytope

A reflexive polytope is a full-dimensional lattice polytope containing the origin in its interior whose polar

$P^*={y in R^d:<y,x>>=-1 for all x in P},$

is also a lattice polytope (Batyrev 1994). Here the point lattice is identified with . Polarity exchanges vertices and facets and preserves reflexivity.

Nill (2005) conjectured that a -dimensional reflexive polytope has at most vertices. Hofscheier and Nill (2026) constructed an eight-dimensional reflexive polytope with 1371 facets, so its polar has more than vertices. They refuted the bound in every dimension . For each fixed and all sufficiently large , their construction gives at least facets for a positive constant . The authors credit GPT-6.1 Sol with finding the counterexamples and developing the general proof.

References

Batyrev, V. V. "Dual Polyhedra and Mirror Symmetry for Calabi-Yau Hypersurfaces in Toric Varieties." J. Algebraic Geom. 3, 493-535, 1994. https://arxiv.org/abs/alg-geom/9310003.

Hofscheier, J. and Nill, B. "Reflexive Polytopes Can Have Doubly Exponentially Many Vertices." 6 Nov 2026. https://arxiv.org/abs/2611.08819v1.

Nill, B. "Gorenstein Toric Fano Varieties." Manuscripta Math. 116, 183-210, 2005. https://doi.org/10.1007/s00229-004-0532-3.

Reflexive Polytope

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