Few distinct distances force a thin annulus about the centroid

Let $n\ge 1$ and let $p=(p_1,\dots,p_n)$ be an injective configuration of $n$ points in the plane, with image $P=\{p_1,\dots,p_n\}$. Assume that no four points of $P$ lie on a common circle. Let $R$ be the set of distances $|x-y|$ with $x,y\in P$ and $x\ne y$, and let $D(P)=|R|$ be the number of distinct distances determined by $P$. Put

If $\delta$ is a real number with $D(P)\le(1/3+\delta)\,n$, then every index $i$ satisfies

A circle about a point of $P$ contains at most three other points of $P$, so $3D(P)\ge n-1$. The theorem is a stability statement for this bound: when $D(P)$ is at most $(1/3+\delta)n$, all points lie in an annulus about the centroid of relative squared width $12\delta+4/n$. No hypothesis on collinear points is used.

Formalization Note Here $c$ is configCentroid p, $M$ is centroidSquaredMass p, $\rho^2$ is distanceAnnulusRadiusSq p, and $R$ is OrderedDistanceValues p, the set of values $|p_i-p_j|$ with $i\ne j$, which is the distance set of $P$ because $p$ is injective. $D(P)$ is NumDistances p. "On a common circle" is Mathlib's Cospherical in the Euclidean plane, applied to four-element subsets of $P$.