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arXiv · 1507.01631

A generalised isodiametric problem

Christos Pelekis
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Abstract

Fix positive integers $a$ and $b$ such that $a> b\geq 2$ and a positive real $\delta>0$. Let $S$ be a planar set of diameter $\delta$ having the following property: for every $a$ points in $S$, at least $b$ of them have pairwise distances that are all less than or equal to $2$. What is the maximum Lebesgue measure of $S$? In this paper we investigate this problem. We discuss the, devious, motivation that leads to its formulation and provide upper bounds on the Lebesgue measure of $S$. Our main result is based on a generalisation of a theorem that is due to Heinrich Jung. In certain instances we are able to find the extremal set but the general case seems elusive.

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Christos Pelekis. 2015-07-06. A generalised isodiametric problem. https://arxiv.org/abs/1507.01631

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