A convex set with a rich difference

Abstract

We construct a convex set $A$ with cardinality $2n$ and with the property that an element of the difference set $A-A$ can be represented in $n$ different ways. We also show that this construction is optimal by proving that for any convex set $A$, the maximum possible number of representations an element of $A-A$ can have is $\lfloor |A|/2\rfloor$.

This version has been edited to include a reference to a paper of Schoen, which we were not previously aware of. The paper of Schoen proves a similar result, with a worse multiplicative constant