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Homogeneous maximizers of the Blaschke--Santalo-type functionals

Alexander V. Kolesnikov

TL;DR

This work extends Blaschke–Santaló type inequalities to $N\ge 2$ copies with a general homogeneous cost $c$ and symmetric convex sets, establishing a bridge between set and functional formulations via optimal transport duality. It proves that, under natural homogeneity and convexity assumptions, maximizers $V_i$ are homogeneous and identifies explicit consistency conditions linking the cost, densities, and homogeneity parameters, thereby deriving a homogeneous structure for the extremizers. A generalized Steiner-type symmetrization is developed to increase the Blaschke–Santaló functional and reduce to symmetric/unconditional configurations, while a key main example demonstrates that, for a product-like cost, maximizers are given by Minkowski functionals of maximizing sets, yielding explicit homogeneous extremizers. The results deepen connections between multimarginal optimal transport, Monge–Ampère theory on spheres, and the Gaussian saturation paradigm, with implications for sharp transportation–information inequalities and potential extensions to non-Gaussian reference measures.

Abstract

We study Blaschke--Santal{ó}-type inequalities for $N \ge 2$ sets (functions) and a special class of cost functions. In particular, we prove new results about reduction of the maximization problem for the Blaschke--Santal{ó}-type functional to homogeneous case (functional inequalities on the sphere) and extend the symmetrization argument to the case of $N > 2$ sets. We also discuss links to the multimagrinal optimal transportation problem and the related sharp transportation-information inequalities.

Homogeneous maximizers of the Blaschke--Santalo-type functionals

TL;DR

This work extends Blaschke–Santaló type inequalities to copies with a general homogeneous cost and symmetric convex sets, establishing a bridge between set and functional formulations via optimal transport duality. It proves that, under natural homogeneity and convexity assumptions, maximizers are homogeneous and identifies explicit consistency conditions linking the cost, densities, and homogeneity parameters, thereby deriving a homogeneous structure for the extremizers. A generalized Steiner-type symmetrization is developed to increase the Blaschke–Santaló functional and reduce to symmetric/unconditional configurations, while a key main example demonstrates that, for a product-like cost, maximizers are given by Minkowski functionals of maximizing sets, yielding explicit homogeneous extremizers. The results deepen connections between multimarginal optimal transport, Monge–Ampère theory on spheres, and the Gaussian saturation paradigm, with implications for sharp transportation–information inequalities and potential extensions to non-Gaussian reference measures.

Abstract

We study Blaschke--Santal{ó}-type inequalities for sets (functions) and a special class of cost functions. In particular, we prove new results about reduction of the maximization problem for the Blaschke--Santal{ó}-type functional to homogeneous case (functional inequalities on the sphere) and extend the symmetrization argument to the case of sets. We also discuss links to the multimagrinal optimal transportation problem and the related sharp transportation-information inequalities.
Paper Structure (8 sections, 21 theorems, 137 equations)

This paper contains 8 sections, 21 theorems, 137 equations.

Table of Contents

  1. Introduction
  2. Optimal transportation approach
  3. Homogeneity of the maximizers
  4. The classical Blaschke--Santaló inequality revisited
  5. Inequalities for sets and for functions
  6. Symmetrization
  7. Main example
  8. Concluding remarks and open questions

Key Result

Theorem 1.1

Let $N \ge 2$ be a natural number. Consider the following cost function on $(\mathbb{R}^n)^N$. Assume we are given $N$ symmetric convex sets $A_i \subset \mathbb{R}^n$, $1 \le i \le N$ such that $c(x_1,\cdots,x_N) \le 1$ on the set $\prod_{i=1}^N A_i \subset (\mathbb{R}^n)^N$. Then Let $\{V_i\}$, $1 \le i \le N,$ be even measurable functions with values in $(-\infty,+\infty]$ and satisfying $\su

Theorems & Definitions (46)

  • Theorem 1.1
  • Definition 1.2
  • Theorem 1.3
  • Remark 2.1
  • Theorem 2.2
  • proof
  • Corollary 2.3
  • Theorem 2.4
  • proof
  • Theorem 2.5
  • ...and 36 more