Strict concavity properties of cross covariograms

2025/08/05 by Bianchi, Gabriele, Burchard, Almut, Lin, Lawrence
#52A38 (primary) 26D15 #60D05 (secondary) #FOS: Mathematics #Functional Analysis (math.FA) #Metric Geometry (math.MG)

paper · doi:10.48550/arxiv.2508.03887

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.

Citations

• A Rogers--Brascamp--Lieb--Luttinger inequality in the space of matrices
• A quantitative stability result for the Prékopa-Leindler inequality for arbitrary measurable functions
• A stochastic Prekopa-Leindler inequality for log-concave functions
• Random Measurable Sets and Covariogram Realisability Problems
• Confirmation of Matheron's conjecture on the covariogram of a planar convex body
• The volume of the intersection of a convex body with its translates

Related

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• Confirmation of Matheron's conjecture on the covariogram of a planar convex body
• On convex bodies that are characterizable by volume function
• On the improvement of concavity of convex measures
• A note on the reduction of the slicing problem to centrally symmetric convex bodies