Strict concavity properties of cross covariograms
Metric Geometry
2025-08-07 v1 Functional Analysis
Abstract
It is well-known that the cross covariogram of two convex bodies in n dimensions is 1/n-concave on its support. This paper provides conditions for strict 1/n-concavity in dimension n>1, and an analysis of how it can fail. Among the implications are that (i.) the cross covariogram of strictly convex bodies is strictly 1/n-concave, unless one body contains a translate of the other in its interior, and (ii.) the cross covariogram of an arbitrary convex body with its reflection through the origin is strictly log-concave.
Cite
@article{arxiv.2508.03887,
title = {Strict concavity properties of cross covariograms},
author = {Gabriele Bianchi and Almut Burchard and Lawrence Lin},
journal= {arXiv preprint arXiv:2508.03887},
year = {2025}
}
Comments
14 pages, 3 figures