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A quantitative version of the Steinhaus theorem

Alex Iosevich, Jonathan Pakianathan

Abstract

The classical Steinhaus theorem (\cite{Steinhaus1920}) says that if $A \subset {\Bbb R}^d$ has positive Lebesgue measure than $A-A=\{x-y: x,y \in A\}$ contains an open ball. We obtain some quantitative lower bounds on the size of this ball and in some cases, relate it to natural geometric properties of $\partial A$. We also study the process $K_n =\frac{1}{2}(K_{n-1} - K_{n-1})$ when $K_0$ is a compact subset of $\mathbb{R}^d$ and determine various aspects of its convergence to $Conv(K_1)$, the convex hull of $K_1$. We discuss some connections with convex geometry, Weyl tube formula and the Kakeya needle problem. \noindent {\it Keywords: Measure theory, Steinhaus theorem, Convex geometry, Weyl tube formula.} \noindent 2020 {\it Mathematics Subject Classification:} Primary: 28A75, 52A27. Secondary: 52A30, 53A07.

A quantitative version of the Steinhaus theorem

Abstract

The classical Steinhaus theorem (\cite{Steinhaus1920}) says that if has positive Lebesgue measure than contains an open ball. We obtain some quantitative lower bounds on the size of this ball and in some cases, relate it to natural geometric properties of . We also study the process when is a compact subset of and determine various aspects of its convergence to , the convex hull of . We discuss some connections with convex geometry, Weyl tube formula and the Kakeya needle problem. \noindent {\it Keywords: Measure theory, Steinhaus theorem, Convex geometry, Weyl tube formula.} \noindent 2020 {\it Mathematics Subject Classification:} Primary: 28A75, 52A27. Secondary: 52A30, 53A07.
Paper Structure (7 sections, 6 theorems, 89 equations)

This paper contains 7 sections, 6 theorems, 89 equations.

Table of Contents

  1. Introduction
  2. Some initial observations
  3. The Banach-Steinhaus Process
  4. Banach Steinhaus process in dimension 1
  5. Banach-Steinhaus process in higher dimensions
  6. Kakeya Needle Problem and Blaschke-Santaló diagrams
  7. Proof of Theorem \ref{['steinhausradiusthm']} and Theorem \ref{['mainmodified']}

Key Result

Theorem 1.2

Let $U \subset {\Bbb R}^d$ be of finite positive Lebesgue measure. Let and define Then

Theorems & Definitions (21)

  • Definition 1.1
  • Theorem 1.2
  • Remark 2.1
  • Example 2.2
  • Example 2.3
  • Remark 2.4
  • Theorem 2.5
  • Proposition 3.1
  • proof
  • Example 3.2: 2 points in $\mathbb{R}^1$
  • ...and 11 more