On sumsets and convex hull

TL;DR

This paper explores sumsets and convex hulls, extending classical bounds on sumset sizes and characterizing equality cases, with implications for understanding the structure of finite sets in Euclidean space.

Contribution

It generalizes the Matolcsi-Ruzsa lower bound for sumsets involving convex hulls and characterizes the conditions for equality, using polytope triangulations.

Findings

Generalized the lower bound for |A+kB| when B is d-dimensional

Characterized the equality case of the Matolcsi-Ruzsa bound

Connected sumset properties with polytope triangulations

Abstract

One classical result of Freimann gives the optimal lower bound for the cardinality of A+A if A is a d-dimensional finite set in the Euclidean d-space. Matolcsi and Ruzsa have recently generalized this lower bound to |A+kB| if B is d-dimensional, and A is contained in the convex hull of B. We characterize the equality case of the Matolcsi-Ruzsa bound. The argument is based partially on understanding triangulations of polytopes.