A closed \((n+1)\)-convex set in \({\mathbb{R}}^ 2\) is a union of \(n^ 6\) convex sets (Q919620)
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scientific article; zbMATH DE number 4161569
| Language | Label | Description | Also known as |
|---|---|---|---|
| English | A closed \((n+1)\)-convex set in \({\mathbb{R}}^ 2\) is a union of \(n^ 6\) convex sets |
scientific article; zbMATH DE number 4161569 |
Statements
A closed \((n+1)\)-convex set in \({\mathbb{R}}^ 2\) is a union of \(n^ 6\) convex sets (English)
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1990
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It is shown that if a closed set S in the plane is \(n+1\)-convex, then it has no more than \(n^ 4\) holes. As a consequence, it can be covered by \(\leq n^ 6\) convex subsets. This is an improvement on the bound of \(2^ n\cdot n^ 3\) obtained by \textit{M. Breen} and \textit{D. C. Kay} [Israel J. Math. 24, 217-233 (1976; Zbl 0342.52006)].
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convex sets
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