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

TL;DR

The paper proves a tight upper bound on the number of facets of twinned chain polytopes $\Gamma(P,Q)$ arising from posets, showing $N(\Gamma(P,Q))\le 6^{d/2}$ for even dimension $d$ (and a weaker bound for odd $d$), thereby partially confirming Nill's conjecture within this natural non-centrally symmetric class. The authors derive a facet-counting formula via maximal chains of ordinal sums and exploit direct-sum decompositions, along with detailed degree analyses of comparability graphs and inductive arguments. Equality is completely characterized: it occurs precisely when $P$ and $Q$ are direct sums of $d/2$ copies of the two-element antichain, with a poset-graph isomorphism linking their comparability structures. The work connects combinatorial poset structure to geometric facet structure, contributing to the broader understanding of reflexive polytopes and their facet counts, and highlighting the role of del Pezzo-like free sums in attaining the bound.

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 $6^{d/2}$.

Figures (3)

• Figure 1: Posets $P$ and $Q$ for which the equality holds in Theorem \ref{['main theorem']}
• Figure 2: Pairs of posets $P, Q \cong {\bf 1} \oplus {\bf I}_2$ in Example \ref{['rei']}
• Figure 3: Twinned chain polytopes for $d=2$

Theorems & Definitions (28)

• Conjecture 1.1: Nillconj
• Theorem 1.2
• Proposition 2.1: T
• Remark 2.2
• Example 2.3
• Proposition 2.4
• proof
• Proposition 2.5
• proof
• Corollary 2.6