A Brunn-Minkowski Theory for the Distribution of Random Pairs of Points in Convex Bodies

Let $K\subset {\mathbb{R}}^n$ be a convex body, let $X$ and $Y$ be independent uniform points of $K$, and set $R_K=(\mathrm{\mid }K\mathrm{\mid }\mathrm{/}{\omega }_n{)}^{1\mathrm{/}n}$. We consider the distribution function $$D_K(\rho )=\mathbb{P}\{\mathrm{\mid }X-Y\mathrm{\mid }\mathrm{/}{R}_K<\rho \},$$ and its homogeneous counterpart \begin{equation*} \mathcal J_ρ(K)=\iint_{K\times K}$\mathbf{1}$_{{|x-y|<ρR_K}}dxdy. \end{equation*} Thus $D_K(\rho )$ is the proportion of ordered pairs of points of $K$ whose distance is less than $\rho R_K$, or equivalently the radial distribution function of the normalized covariogram of $K$. The functional satisfies a Brunn–Minkowski inequality, with equality precisely for homothetic bodies, and its differential gives a first Minkowski inequality. We characterize the Borel measures that occur as Wulff differentials: they are exactly the nonzero finite positive measures with zero centroid which are not concentrated on a great subsphere, and the corresponding body is unique up to translation. We further prove fixed-volume and stationary rigidity of the ball in the subcritical range, together with regularity for the resulting equation. At saturation, the equation is the classical Minkowski equation.