Weighted Brunn-Minkowski Theory I: On Weighted Surface Area Measures

Abstract

The Brunn-Minkowski theory in convex geometry concerns, among other things, the volumes, mixed volumes, and surface area measures of convex bodies. We study generalizations of these concepts to Borel measures with density in ${\mathbb{R}}^n$-- in particular, the weighted versions of mixed volumes (the so-called mixed measures) when dealing with up to three distinct convex bodies. We then formulate and analyze weighted versions of classical surface area measures, and obtain a new integral formula for the mixed measure of three bodies. As an application, we prove a Bézout-type inequality for rotational invariant log-concave measures, generalizing a result by Artstein-Avidan, Florentin and Ostrover. The results are new and interesting even for the special case of the standard Gaussian measure.

arXiv comment30 pages, Keywords: Brunn-Minkowski theory, surface area, Gaussian measure, zonoids, mixed volumes, mixed measures. Previously titled "Weighted surface area measures and Brunn-Minkowski theory", the paper was split and improved upon

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Cited by in corpus (6)

• Weighted Brunn-Minkowski Theory I: On Weighted Surface Area Measures
• Weighted Minkowski's Existence Theorem and Projection Bodies
• On the volume of the Minkowski sum of zonoids
• Volumes of subset Minkowski sums and the Lyusternik region
• The (Self-Similar, Variational) Rolling Stones
• Weighted Brunn-Minkowski Theory II: Inequalities for Mixed Measures and Applications