If $C,D\subseteq {\mathbb{R}}^2$ then the distance between $C$ and $D$ is defined by$$\delta (C,D)=\underset{\begin{array}{c}c\in C\\ d\in D\end{array}}{\inf }\Vert c-d\Vert .$$Let $h(n)$ be the maximal number of unit distances between disjoint convex translates. That is, the maximal $m$ such that there is a compact convex set $C\subset {\mathbb{R}}^2$ and a set $X$ of size $n$ such that all $(C+x{)}_{x\in X}$ are disjoint and there are $m$ pairs $x_1,x_2\in X$ such that$$\delta (C+x_1,C+x_2)=1.$$Determine $h(n)$ - in particular, prove that there exists a constant $c>0$ such that $h(n)>n^{1+c}$ for all large $n$.