Large convex sets in difference sets

We give a construction of a convex set A subset of R$A \subset \mathbb {R}$ with cardinality n$n$ such that A-A$A-A$ contains a convex subset with cardinality Omega(n2)$\Omega (n<^>2)$. We also consider the following variant of this problem: given a convex set A$A$, what is the size of the largest matching M subset of AxA$M \subset A \times A$ such that the set {a-b:(a,b)is an element of M}$$\begin{equation*} \lbrace a-b: (a,b) \in M \rbrace \end{equation*}$$is convex? We prove that there always exists such an M$M$ with |M|>= n$|M| \geqslant \sqrt n$, and that this lower bound is best possible, up to a multiplicative constant.