Erdős Problem #94 — Squared Distance Multiplicities in Convex Point Sets

Let $P$ be a finite set of points in $\mathbb{R}^2$ whose points are in convex position. Let $\{u_1,\ldots,u_t\}$ be the set of distances determined by pairs of points of $P$, and let $f(u_i)$ be the number of pairs of points of $P$ at distance $u_i$. Prove that there is a constant $C>0$ such that, for every such $P$,

$$\sum_{i=1}^t f(u_i)^2 \le C|P|^3.$$