A convex set with a rich difference

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 $\lfloor \mathrm{\mid }A\mathrm{\mid }\mathrm{/}2\rfloor$.