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arXiv · 2508.03887

Strict concavity properties of cross covariograms

Gabriele Bianchi·Almut Burchard·Lawrence Lin
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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.

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Gabriele Bianchi, Almut Burchard, Lawrence Lin. 2025-08-05. Strict concavity properties of cross covariograms. https://arxiv.org/abs/2508.03887

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