An improved bound for sumsets of thick compact sets via the Shapley--Folkman theorem
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866913008063610880 |
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| author | Kominers, Scott Duke |
| author_facet | Kominers, Scott Duke |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_04889 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | An improved bound for sumsets of thick compact sets via the Shapley--Folkman theorem Kominers, Scott Duke Metric Geometry Classical Analysis and ODEs Combinatorics 52A20 (primary), 28A80, 52A27, 28A75 (secondary), 91B50 (tertiary) 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. |
| title | An improved bound for sumsets of thick compact sets via the Shapley--Folkman theorem |
| topic | Metric Geometry Classical Analysis and ODEs Combinatorics 52A20 (primary), 28A80, 52A27, 28A75 (secondary), 91B50 (tertiary) |
| url | https://arxiv.org/abs/2604.04889 |