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A new infinitesimal form of the Prékopa--Leindler inequality with multiplicative structure and applications

Sotiris Armeniakos, Jacopo Ulivelli

TL;DR

This work derives a new infinitesimal form of the Prékopa–Leindler inequality with a multiplicative structure that simultaneously yields, and sharpens, boundary Poincaré and Brascamp–Lieb inequalities. The authors develop a perturbative Wulff-shape framework and a functional Fenchel–Legendre perspective to obtain a unified, multiplicative inequality expressed via three bilinear forms, and prove a Cauchy–Schwarz-type bound that improves existing inequalities. Building on this, they establish a stability result for the weighted Poincaré inequality, extend the theory to Sobolev spaces, and formulate a novel variational approach to the dimensional Brunn–Minkowski conjecture by solving a weighted elliptic PDE on the boundary of convex bodies. The paper further analyzes the symmetric, Hessian-pinched regime, proving coercivity, existence/uniqueness of minimizers, and deriving explicit bounds on the concavity power $p(ackslash mu,K)$ in terms of dimension and curvature data, thereby offering new insights and techniques beyond the classical $L_2$ method. Overall, the results connect convex-geometric perturbations, PDE techniques, and variational principles to address stability and dimensional concavity questions for log-concave measures.

Abstract

By differentiating a concavity principle arising from the Prékopa--Leindler inequality, we obtain a statement simultaneously strengthening the weighted boundary Poincaré inequality and the Brascamp--Lieb variance inequality. The resulting inequality possesses a multiplicative structure, which we exploit to develop an alternative to the (by now classical) $L_2$ method in the study of geometric and analytic inequalities. We apply this approach to derive a stability estimate for the weighted Poincaré inequality and to investigate the dimensional Brunn--Minkowski conjecture. In particular, in the latter setting, we obtain new reformulations together with several partial results.

A new infinitesimal form of the Prékopa--Leindler inequality with multiplicative structure and applications

TL;DR

This work derives a new infinitesimal form of the Prékopa–Leindler inequality with a multiplicative structure that simultaneously yields, and sharpens, boundary Poincaré and Brascamp–Lieb inequalities. The authors develop a perturbative Wulff-shape framework and a functional Fenchel–Legendre perspective to obtain a unified, multiplicative inequality expressed via three bilinear forms, and prove a Cauchy–Schwarz-type bound that improves existing inequalities. Building on this, they establish a stability result for the weighted Poincaré inequality, extend the theory to Sobolev spaces, and formulate a novel variational approach to the dimensional Brunn–Minkowski conjecture by solving a weighted elliptic PDE on the boundary of convex bodies. The paper further analyzes the symmetric, Hessian-pinched regime, proving coercivity, existence/uniqueness of minimizers, and deriving explicit bounds on the concavity power in terms of dimension and curvature data, thereby offering new insights and techniques beyond the classical method. Overall, the results connect convex-geometric perturbations, PDE techniques, and variational principles to address stability and dimensional concavity questions for log-concave measures.

Abstract

By differentiating a concavity principle arising from the Prékopa--Leindler inequality, we obtain a statement simultaneously strengthening the weighted boundary Poincaré inequality and the Brascamp--Lieb variance inequality. The resulting inequality possesses a multiplicative structure, which we exploit to develop an alternative to the (by now classical) method in the study of geometric and analytic inequalities. We apply this approach to derive a stability estimate for the weighted Poincaré inequality and to investigate the dimensional Brunn--Minkowski conjecture. In particular, in the latter setting, we obtain new reformulations together with several partial results.
Paper Structure (11 sections, 25 theorems, 180 equations)

This paper contains 11 sections, 25 theorems, 180 equations.

Table of Contents

  1. Introduction
  2. A new infinitesimal form of the Prékopa--Leindler inequality.
  3. Stability estimates and the dimensional Brunn--Minkowski conjecture.
  4. Structure of the paper.
  5. A concavity principle along functional Wulff shapes
  6. Stability for the Poincaré inequality
  7. Concavity powers of strictly log-concave measures
  8. Existence of a minimizer and the Lax--Milgram theorem.
  9. A reformulation of the dimensional Brunn--Minkowski conjecture.
  10. Symmetric case and Hessian pinching
  11. Acknowledgments.

Key Result

Theorem 1.1

Consider a non-empty compact convex set $K \subset \mathbb{R}^n$ such that $\partial K$ is a strictly convex manifold of class $C^2$, and a convex function $u \in C^2(\mathbb{R}^n)$ such that its Hessian matrix is positive definite. Then for every Lipschitz function $\rho: \partial K \to \mathbb{R}$ holds.

Theorems & Definitions (44)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Lemma 2.1
  • proof
  • Proposition 2.2
  • proof
  • Lemma 2.3
  • proof
  • ...and 34 more