  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.
