Large Convex sets in Difference sets

Abstract: We give a construction of a convex set $A \subset \mathbb R$ with cardinality $n$ such that $A-A$ contains a convex subset with cardinality $\Omega (n^2)$. We also consider the following variant of this problem: given a convex set $A$, what is the size of the largest matching $M \subset A \times A$ such that the set [ { a-b : (a,b) \in M } ] is convex? We prove that there always exists such an $M$ with $|M| \geq \sqrt n$, and that this lower bound is best possible, up a multiplicative constant.