If $C,D\subseteq \mathbb{R}^2$ then the distance between $C$ and $D$ is defined by

$$\delta(C,D)=\inf_{\substack{c\in C\\ d\in D}}\| c-d\|.$$

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 $n$.