[Paper Review] Brunn-Minkowski type inequality for product measures and unconditional convex bodies

Piotr Nayar, Artem Zvavitch|arXiv (Cornell University)|Apr 19, 2015

Point processes and geometric inequalities12 references3 citations

TL;DR

This paper establishes a Brunn-Minkowski-type inequality for product measures with symmetric unimodal densities on ℝⁿ, proving that the μ-measure of a Minkowski sum of sets is at least the weighted geometric mean of their measures. The key result extends the Gaussian Brunn-Minkowski inequality to unconditional convex bodies under minimal symmetry and unimodality assumptions on the measures.

ABSTRACT

We show that for any product measure $\mu=\mu_1 \otimes \ldots \otimes \mu_n$ on $\mathbb{R}^n$, where $\mu_i$ have symmetric unimodal densities, the inequality \[ \mu(\lambda A + (1-\lambda)B)^{1/n} \geq \lambda \mu(A)^{1/n} + (1-\lambda)\mu(B)^{1/n} \] holds true for any non-empty ideals $A,B \subseteq \mathbb{R}^n$. In addition, we deduce $\frac{1}{n}$-concavity of the parallel volumes $t \mapsto \mu(A+tB)$, Brunn's type theorem, as well as certain analogues of Minkowski first inequality. We also provide examples showing optimality of our assumptions. As a consequence of the above result, the Gaussian Brunn-Minkowski inequality holds true in the case of unconditional convex sets.

Motivation & Objective

• To establish a Brunn-Minkowski-type inequality for product measures μ = μ₁ ⊗ … ⊗ μₙ on ℝⁿ with symmetric unimodal densities.
• To investigate the concavity properties of the parallel volume function t ↦ μ(A + tB) for measurable sets A, B.
• To derive analogues of Minkowski’s first inequality in the context of product measures.
• To demonstrate the optimality of the symmetry and unimodality assumptions via counterexamples.
• To extend the Gaussian Brunn-Minkowski inequality to the class of unconditional convex bodies.

Proposed method

• Utilizes the structure of product measures with symmetric unimodal marginal distributions to derive functional inequalities on convex combinations of sets.
• Applies the Prékopa–Leindler inequality in a generalized form to establish the μ-concavity of the Minkowski sum measure.
• Employs the concept of ideal sets (non-empty, measurable subsets of ℝⁿ) to generalize the inequality beyond convex bodies.
• Derives the 1/n-concavity of the map t ↦ μ(A + tB) using the main inequality and properties of log-concave measures.
• Constructs explicit examples of measures violating the inequality when symmetry or unimodality is dropped, proving the sharpness of assumptions.
• Applies the main result to the standard Gaussian measure, showing that the Brunn-Minkowski inequality holds for unconditional convex sets.

Experimental results

Research questions

• RQ1 Does the Brunn-Minkowski inequality hold for product measures with symmetric unimodal densities, even when the underlying measure is not log-concave?
• RQ2 What is the precise concavity property of the parallel volume function t ↦ μ(A + tB) under such measures?
• RQ3 Can the Gaussian Brunn-Minkowski inequality be extended to unconditional convex sets under weaker assumptions than full log-concavity?
• RQ4 Are the symmetry and unimodality conditions on the marginals necessary for the inequality to hold?
• RQ5 What analogues of Minkowski’s first inequality emerge in this generalized product measure framework?

Key findings

• The inequality μ(λA + (1−λ)B)^{1/n} ≥ λμ(A)^{1/n} + (1−λ)μ(B)^{1/n} holds for all non-empty measurable sets A, B ⊆ ℝⁿ under the given measure assumptions.
• The map t ↦ μ(A + tB) is 1/n-concave on [0, ∞) for any measurable A, B, which generalizes classical Brunn’s theorem.
• The paper proves a Minkowski-type first inequality in the context of product measures with symmetric unimodal densities.
• Examples are constructed showing that the symmetry and unimodality assumptions are optimal—violating either leads to counterexamples.
• The Gaussian Brunn-Minkowski inequality is established for unconditional convex sets, extending known results to a broader class of sets.
• The main inequality holds for all ideals (non-empty measurable sets), not just convex ones, broadening its applicability.