Strict concavity properties of cross covariograms

TL;DR

This paper investigates the strict concavity properties of cross covariograms of convex bodies, establishing conditions for strict 1/n-concavity and analyzing cases of failure, with implications for convex geometry.

Contribution

It provides new conditions for strict 1/n-concavity of cross covariograms in higher dimensions and analyzes when this property can fail.

Findings

Cross covariogram of strictly convex bodies is strictly 1/n-concave unless one contains a translate of the other.

Cross covariogram of a convex body with its reflection is strictly log-concave.

Conditions for strict concavity properties in convex geometry are characterized.

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.