An Upper Bound Theorem concerning lattice polytopes

TL;DR

This paper extends Stanley’s proof of the Upper Bound Conjecture to Ehrhart rings, providing new volume bounds and inequalities for delta-vectors of integrally closed lattice polytopes, with applications to reflexive and order polytopes.

Contribution

It introduces upper bounds and inequalities for integrally closed lattice polytopes, expanding the understanding of their combinatorial and geometric properties.

Findings

Derived upper bounds for the volume of integrally closed lattice polytopes.

Established inequalities for the delta-vector of such polytopes.

Applied results to reflexive and order polytopes.

Abstract

R. P. Stanley proved the Upper Bound Conjecture in 1975. We imitate his proof for the Ehrhart rings. We give some upper bounds for the volume of integrally closed lattice polytopes. We derive some inequalities for the delta-vector of integrally closed lattice polytopes. Finally we apply our results for reflexive integrally closed and order polytopes.