A convex set with a rich difference

TL;DR

This paper constructs a convex set with a specific difference set property and proves that this construction is optimal, establishing a maximum number of representations for elements in the difference set.

Contribution

It introduces a convex set with a rich difference property and proves the optimality of this construction in terms of representation counts.

Findings

Constructed a convex set with a difference set where elements have multiple representations.

Proved that the maximum number of representations for any element in the difference set is half the size of the set.

Established the optimality of the construction with a proven upper bound.

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 $.