Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Scott Duke Kominers

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.

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

Abstract

Let be compact sets of positive diameter with Feng--Wu thickness at least . Feng and Wu proved that has non-empty interior when . We show that already suffices. In particular, since , the bound is enough. For fixed dimension , this improves the exponent in from to , while introducing only an explicit factor of . 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.

Paper Structure

This paper contains 11 sections, 10 theorems, 122 equations.

Table of Contents

  1. Introduction
  2. Key ingredients
  3. A support-function perturbation lemma
  4. Finite discretization of thickness
  5. The Shapley--Folkman theorem in radius form
  6. Absorption of residuals
  7. A parameterized quantitative thick-sum theorem
  8. Optimization and proof of the main theorem
  9. Concluding remarks
  10. The mechanism behind the improvement
  11. Sharpness and dimension-dependence

Key Result

Theorem 1

Let $E_1,\dots,E_n\subset \mathbb{R}^d$ be compact sets with $\mathop{\mathrm{diam}}\nolimits(E_i)>0$ and If we have then $E_1+\cdots+E_n$ has non-empty interior. In particular, since $0<c\le 1$, suffices. $\blacktriangleleft$$\blacktriangleleft$

Theorems & Definitions (25)

  • Theorem 1
  • Remark 2: Positive-diameter hypothesis
  • Remark 3: Comparison with Feng--Wu
  • Lemma 4
  • proof
  • Lemma 5
  • proof
  • Lemma 6: Conic Carathéodory
  • proof
  • Lemma 7: Shapley--Folkman, classical form
  • ...and 15 more