Facet numbers of non-centrally symmetric reflexive polytopes arising from posets

TL;DR

This paper investigates the facet counts of a class of non-centrally symmetric reflexive polytopes derived from posets, establishing an upper bound and characterizing cases of equality related to del Pezzo polygons.

Contribution

It proves an upper bound on the number of facets of twinned chain polytopes and characterizes when this bound is attained, partially confirming Nill's conjecture.

Findings

Maximum facet count is 6^{d/2} for d-dimensional twinned chain polytopes.

Equality occurs iff the polytope is a free sum of del Pezzo polygons.

Supports Nill's conjecture on reflexive polytope facets.

Abstract

Twinned chain polytopes form a broad class of non-centrally symmetric reflexive polytopes and exhibit intriguing structures. In the present paper, we show that the number of facets of $d$-dimensional twinned chain polytopes is at most $6^{d/2}$. In case $d$ is even, the equality holds if and only if the polytope is isomorphic to a free sum of $d/2$ copies of del Pezzo polygons. This result contributes a partial answer to Nill's conjecture: the number of facets of a $d$-dimensional reflexive polytope is at most $6^{d/2}$.