An improved bound for sumsets of thick compact sets via the Shapley--Folkman theorem

TL;DR

This paper improves bounds on the number of summands needed for sumsets of thick compact sets in Euclidean space to have non-empty interior, using a convexification approach based on the Shapley--Folkman theorem.

Contribution

It introduces a new bound that reduces the exponent in the thickness parameter from 3 to 2, with an explicit dependence on the dimension.

Findings

The new bound is n > 6√d c^{-2} for sumsets to have non-empty interior.

The proof replaces a stepwise enlargement with a simultaneous convexification.

The bound explicitly depends on the dimension d and the thickness c.

Abstract

Let $E_1,\dots,E_n \subset \mathbb{R}^d$ be compact sets of positive diameter with Feng--Wu thickness at least $c>0$. Feng and Wu proved that $E_1+\cdots+E_n$ has non-empty interior when $n>2^{11}c^{-3}+1$. We show that \[n>\frac{\sqrt d}{(\sqrt{1+c}-1)^2}=\frac{\sqrt d\,(\sqrt{1+c}+1)^2}{c^2}\] already suffices. In particular, since $0<c\le 1$, the bound $n>6\sqrt d\,c^{-2}$ is enough. For fixed dimension $d$, this improves the exponent in $c^{-1}$ from $3$ to $2$, while introducing only an explicit factor of $\sqrt d$. The proof replaces the one-summand-at-a-time enlargement of Feng--Wu by a simultaneous convexification step based on a radius form of the Shapley--Folkman theorem.