Sumsets and the convex hull

TL;DR

This paper extends Freiman's inequality to higher dimensions, exploring how the sumset size relates to convex hull inclusion and multiple additions of the set.

Contribution

It introduces new bounds and relations for sumsets in higher dimensions considering convex hull inclusion and multiple set additions.

Findings

Extended Freiman's inequality to higher dimensions.

Established bounds for sumsets with convex hull inclusion.

Analyzed sumset sizes under multiple set additions.

Abstract

We extend Freiman's inequality on the cardinality of the sumset of a $d$ dimensional set. We consider different sets related by an inclusion of their convex hull, and one of them added possibly several times.