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

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.