A Brunn-Minkowski Theory for the Distribution of Random Pairs of Points in Convex Bodies
Abstract
Let K⊂ℝn be a convex body, let X and Y be independent uniform points of K, and set RK=(|K|/ωn)1/n. We consider the distribution function DK(ρ)=ℙ{|X−Y|/RK<ρ}, and its homogeneous counterpart 𝒥ρ(K)=iintK× K1{|x−y|<ρRK}dxdy. Thus DK(ρ) is the proportion of ordered pairs of points of K whose distance is less than ρRK, 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.
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BibTeX
@misc{wang2026brunnminkowski,
title = {{A Brunn-Minkowski Theory for the Distribution of Random Pairs of Points in Convex Bodies}},
author = {Sinan Wang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.01916},
eprint = {2610.01916},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.01916},
}
RIS
TY - UNPB
TI - A Brunn-Minkowski Theory for the Distribution of Random Pairs of Points in Convex Bodies
AU - Sinan Wang
PY - 2026
DA - 2026/10/01
PB - arXiv
DO - 10.48550/arxiv.2610.01916
UR - https://arxiv.org/abs/2610.01916
AB - Let K⊂ℝ^n be a convex body, let X and Y be independent uniform points of K, and set R_K=(|K|/ω_n)^1/n. We consider the distribution function D_K(ρ)=ℙ{|X−Y|/R_K<ρ}, and its homogeneous counterpart 𝒥_ρ(K)=iint_K× K1_{|x−y|<ρR_K}dxdy. Thus D_K(ρ) is the proportion of ordered pairs of points of K whose distance is less than ρR_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.
ER -