Nill's facet bound for reflexive lattice polytopes
A full-dimensional lattice polytope $P \subset \mathbb{R}^d$ is reflexive if $0 \in \operatorname{int} P$ and its polar $$P^* = \{y \in \mathbb{R}^d : \langle x, y \rangle \le 1 \text{ for every } x \in P\}$$ is also a lattice polytope. Write $f_{d-1}(P)$ for the number of facets of $P$. Nill proposed the bound below for the number of vertices, which is the same question by polarity. Mori, Mori and Ohsugi proved it for twinned chain polytopes, among which, for even $d$, equality holds exactly for free sums of $d/2$ del Pezzo polygons.
Problem. Does every reflexive lattice polytope $P \subset \mathbb{R}^d$ satisfy $$f_{d-1}(P) \le 6^{d/2}?$$ No simplicity, simpliciality or symmetry assumption is imposed.
Source: Nill (2005), Conjecture 5.2, as a bound on the number of vertices; in this facet form, Mori, Mori and Ohsugi (2026), Conjecture 1.1.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- CO — Combinatorics
- Secondary
- AG, MG
- Keywords
- AIM Open Applied Problems, facet numbers, lattice polytopes, reflexive polytopes
References
@article{mori2026,
author = {Mori, Aki and Mori, Kenta and Ohsugi, Hidefumi},
title = {Facet Numbers of Non-Centrally Symmetric Reflexive Polytopes Arising from Posets},
journal = {Discrete Comput. Geom.},
year = {2026},
doi = {10.1007/s00454-026-00879-x},
eprint = {2511.22981},
archivePrefix = {arXiv},
}
@article{nill2005,
author = {Nill, Benjamin},
title = {Gorenstein toric {F}ano varieties},
journal = {Manuscripta Math.},
volume = {116},
number = {2},
year = {2005},
pages = {183--210},
doi = {10.1007/s00229-004-0532-3},
eprint = {math/0405448},
archivePrefix = {arXiv},
}
@misc{aim-reflexive-polytope-facet-bound,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Nill's facet bound for reflexive lattice polytopes}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/24edaadc6bb272737d8da0745c662c992db07b22/problems/610-reflexive-polytope-facet-bound.md},
note = {Entry 610 at commit 24edaad, 2026-10-15},
}
Resolution claims
A lattice polytope is reflexive if it contains the origin in its interior and its polar is also a lattice polytope. We construct a $45$-dimensional reflexive lattice polytope with $2^{32}\binom{40}{8}+5$ facets, exceeding $6^{45/2}$. This disproves the unrestricted facet bound conjectured by Nill. The construction starts with a lattice polytope having many vertices and a polar whose denominators divide $32$. Five …
Status history
| When | Change | By | Reason |
|---|---|---|---|
| 2026-10-18 | Needs verification → Solution claimed | — | Recorded here: AIM @ 24edaad, paper/manuscript.pdf. |
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00831, "Nill's facet bound for reflexive lattice polytopes", clopen.solutions.