A quantitative version of the Steinhaus theorem
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
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2024
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| _version_ | 1866909079386980352 |
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| author | Iosevich, Alex Pakianathan, Jonathan |
| author_facet | Iosevich, Alex Pakianathan, Jonathan |
| contents | 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.
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{\it Keywords: Measure theory, Steinhaus theorem, Convex geometry, Weyl tube formula.}
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2020 {\it Mathematics Subject Classification:} Primary: 28A75, 52A27. Secondary: 52A30, 53A07. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_12112 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A quantitative version of the Steinhaus theorem Iosevich, Alex Pakianathan, Jonathan Classical Analysis and ODEs 28A75, 52A27, 52A30, 53A07 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. |
| title | A quantitative version of the Steinhaus theorem |
| topic | Classical Analysis and ODEs 28A75, 52A27, 52A30, 53A07 |
| url | https://arxiv.org/abs/2401.12112 |