Non-normal very ample polytopes and their holes

TL;DR

This paper constructs non-normal, very ample convex polytopes of any given dimension with a specified number of holes, advancing the understanding of their geometric and algebraic properties.

Contribution

It demonstrates the existence of non-normal, very ample polytopes with an arbitrary number of holes in any dimension three or higher.

Findings

Existence of non-normal, very ample polytopes with h holes for any h ≥ 1

Construction method applicable in any dimension d ≥ 3

Provides new insights into the structure of holes in convex polytopes

Abstract

In this paper, we show that for given integers $h$ and $d$ with $h \geq 1$ and $d \geq 3$, there exists a non-normal very ample integral convex polytope of dimension $d$ which has exactly $h$ holes.