Suppose $n$ points in $\mathbb{R}^2$ determine a convex polygon and the set of distances between them is $\{u_1,\ldots,u_t\}$. Suppose $u_i$ appears as the distance between $f(u_i)$ many pairs of points. Then $$\sum_i f(u_i)^2 \ll n^3.$$

For every finite planar point set P, the sum over its distinct determined distances of the unordered-pair distance multiplicities equals P.card.choose 2