Convex-normal (pairs of) polytopes

TL;DR

This paper investigates convex-normal polytopes, establishing equivalences between k- and (k+1)-convex-normality for lattice polytopes and extending the concept to pairs of polytopes, with implications for integer decomposition properties.

Contribution

It proves that for lattice polytopes, k- and (k+1)-convex-normality are equivalent for k >= 3 and extends convex-normality to pairs of polytopes with a new decomposition result.

Findings

Equivalence of k- and (k+1)-convex-normality for lattice polytopes at k >= 3

Improved bound for convex-normality from 4d(d+1) to 2d(d+1)

Decomposition property for pairs of rational polytopes with edge length ratios

Abstract

In 2012 Gubeladze (Adv.\ Math.\ 2012) introduced the notion of k-convex-normal polytopes to show that integral polytopes all of whose edges are longer than 4d(d+1) have the integer decomposition property. In the first part of this paper we show that for lattice polytopes there is no difference between k- and (k+1)-convex-normality (for k >= 3) and improve the bound to 2d(d+1). In the second part we extend the definition to pairs of polytopes and show that for rational polytopes P and Q, where the normal fan of P is a refinement of the normal fan of Q, if every edge e_P of P is at least d times as long as the corresponding edge e_Q of Q, then (P+Q) \cap \Z^d = (P\cap \Z^d) + (Q \cap \Z^d).