# Equivalence of Lattice Orbit Polytopes

## TL;DR

This paper introduces a new equivalence relation called normalizer equivalence for core points in permutation group actions, analyzing its properties and implications for infinite classes and optimization problems.

## Contribution

It proposes the concept of normalizer equivalence for core points, expanding the understanding of their classification and finiteness properties under certain group conditions.

## Key findings

- Normalizer equivalence often groups infinitely many vectors together.
- Number of core points up to normalizer equivalence is finite for QI-groups.
- Application shown in simplifying integer convex optimization problems.

## Abstract

Let $G$ be a finite permutation group acting on $\mathbb{R}^d$ by permuting coordinates. A core point (for $G$) is an integral vector $z\in \mathbb{Z}^d$ such that the convex hull of the orbit $Gz$ contains no other integral vectors but those in the orbit $Gz$. Herr, Rehn and Sch\"urmann considered the question for which groups there are infinitely many core points up to translation equivalence, that is, up to translation by vectors fixed by the group. In the present paper, we propose a coarser equivalence relation for core points called normalizer equivalence. These equivalence classes often contain infinitely many vectors up to translation, for example when the group admits an irrational invariant subspace or an invariant irreducible subspace occurring with multiplicity greater than $1$. We also show that the number of core points up to normalizer equivalence is finite if $G$ is a so-called QI-group. These groups include all transitive permutation groups of prime degree. We give an example to show how the concept of normalizer equivalence can be used to simplify integer convex optimization problems.