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Weighted Brunn-Minkowski Theory II: Inequalities for Mixed Measures and Applications

Matthieu Fradelizi, Dylan Langharst, Mokshay Madiman, Artem Zvavitch

Abstract

In "Weighted Brunn-Minkowski Theory I", the prequel to this work, we discussed how recent developments on concavity of measures have laid the foundations of a nascent weighted Brunn-Minkowski theory. In particular, we defined the mixed measures of three convex bodies and obtained its integral representation. In this work, we obtain inequalities for mixed measures, such as a generalization of Fenchel's inequality; this provides a new, simpler proof of the classical volume case. Moreover, we show that mixed measures are connected to the study of log-submodularity and supermodularity of the measure of Minkowski sums of convex bodies. This elaborates on the recent investigations of these properties for the Lebesgue measure. We conclude by establishing that the only Radon measures that are supermodular over the class of compact, convex sets are multiples of the Lebesgue measure. Motivated by this result, we then discuss weaker forms of supermodularity by restricting the class of convex sets.

Weighted Brunn-Minkowski Theory II: Inequalities for Mixed Measures and Applications

Abstract

In "Weighted Brunn-Minkowski Theory I", the prequel to this work, we discussed how recent developments on concavity of measures have laid the foundations of a nascent weighted Brunn-Minkowski theory. In particular, we defined the mixed measures of three convex bodies and obtained its integral representation. In this work, we obtain inequalities for mixed measures, such as a generalization of Fenchel's inequality; this provides a new, simpler proof of the classical volume case. Moreover, we show that mixed measures are connected to the study of log-submodularity and supermodularity of the measure of Minkowski sums of convex bodies. This elaborates on the recent investigations of these properties for the Lebesgue measure. We conclude by establishing that the only Radon measures that are supermodular over the class of compact, convex sets are multiples of the Lebesgue measure. Motivated by this result, we then discuss weaker forms of supermodularity by restricting the class of convex sets.
Paper Structure (15 sections, 35 theorems, 197 equations)

This paper contains 15 sections, 35 theorems, 197 equations.

Table of Contents

  1. Introduction
  2. Inequalities for mixed measures
  3. Minkowski's inequalities
  4. Fenchel-type inequalities
  5. Applications of mixed measures
  6. Supermodularity over convex sets characterizes the Lebesgue Measure
  7. Weaker forms of supermodularity
  8. Direct characterizations of super-or-sub-modularity
  9. Log-submodularity
  10. Appendix A: existence of mixed measures
  11. Appendix B: an inequality for convex functions on ${\mathbb R}$
  12. Establishing the inequality
  13. Optimized form of the inequality
  14. Open question: higher dimensions
  15. Appendix C: supermodular measures must have density

Key Result

Corollary 1.2

Let $K$ and $L$ be symmetric convex bodies in ${\mathbb R}^n, n \geq 2$. Then:

Theorems & Definitions (69)

  • Definition 1.1
  • Corollary 1.2
  • Corollary 1.3
  • Definition 1.4
  • Proposition 1.5: Local form of supermodularity
  • Remark 1.6: Note added in proof
  • Proposition 2.1: Minkowski's first inequality for $F$-concave measures
  • proof
  • Remark 2.2
  • Proposition 2.3: Minkowski's second inequality for $F$-concave measures
  • ...and 59 more