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

Matthieu Fradelizi, Dylan Langharst, Mokshay Madiman, Artem Zvavitch

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.

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 -- 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.
Paper Structure (15 sections, 22 theorems, 151 equations)

This paper contains 15 sections, 22 theorems, 151 equations.

Table of Contents

  1. Introduction
  2. Background
  3. Representation formulae for mixed measures
  4. Weighted versions of Minkowski's inequalities
  5. Bezout-type inequalities and local log-submodularity
  6. Mixed Measures and Weighted Surface Area
  7. Properties of Mixed Measures
  8. Weighted Surface Area and Mixed Measure of two bodies
  9. Integral representation formula for $\mu(A;B,C)$
  10. Weighted versions of Minkowski's first and second inequalities
  11. The Gaussian Measure in the Plane
  12. Local log-submodularity
  13. Inequalities for rotationally invariant log-concave measures
  14. Improved inequalities for a special class of measures
  15. Concluding Remarks

Key Result

Proposition 1.1

Let $L$ be a compact, convex set and $K$ a convex body with the origin in its interior in ${\mathbb R}^n$. Suppose $\mu$ is a Borel measure on ${\mathbb R}^n$ with continuous density. Then,

Theorems & Definitions (42)

  • Proposition 1.1: Representation of Mixed Measures
  • Definition 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Definition 2.1
  • Proposition 2.2
  • proof
  • ...and 32 more