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Higher-Order Reverse Isoperimetric Inequalities for Log-concave Functions

Dylan Langharst, Francisco Marín Sola, Jacopo Ulivelli

TL;DR

This work develops the $m$-th order extension of key affine reverse inequalities from convex geometry to the realm of integrable log-concave functions. Central to the approach is the asymmetric LYZ body $igra f ig floor$, which enables a functional formulation of Zhang’s projection inequality in the $m$-order setting and yields a concise proof via covariogram–LYZ connections. The authors introduce and analyze functional radial mean bodies $R_p^m f$ and Ball bodies $K_p(g)$, including new regimes for $p o(-1,0]$ and $p o ty$, and study their asymptotics, linking them to $ ext{vol}_{nm}(oldsymbol{ ext{Pi}}^{oldsymbol{ullet},m}igra f ig floor)$ and to $ ext{supp}(g_{f,m})=D^m( ext{supp}(f))$. They establish a functional, $m$-th order Rogers–Shephard inequality based on sup-convolution and the functional covariogram, providing sharp equality conditions tied to simplex-type structures in level sets. Overall, the paper broadens the affine-geometry toolkit for function spaces, connects Ball-type and radial-mean body concepts to the log-concave setting, and offers new structural insights with potential applications in high-dimensional convex-analytic problems and related functional inequalities.

Abstract

The Rogers-Shephard and Zhang's projection inequalities are two reverse, affine isoperimetric-type inequalities for convex bodies. Following a classical work by Schneider, both inequalities have been extended to the so-called $m$th-order setting. In this work, we establish the $m$th-order analogues for these inequalities in the setting of log-concave functions. Our proof of the functional Zhang's projection inequality employs properties of the asymmetric LYZ body, significantly streamlining the argument and producing a novel approach for the case $m=1$. Furthermore, we introduce and analyze the radial mean bodies of a log-concave function, thereby providing a functional generalization of Gardner and Zhang's radial mean bodies. These are new even in the case $m=1$. Our development leverages an extension of Ball bodies, which may be of independent interest.

Higher-Order Reverse Isoperimetric Inequalities for Log-concave Functions

TL;DR

This work develops the -th order extension of key affine reverse inequalities from convex geometry to the realm of integrable log-concave functions. Central to the approach is the asymmetric LYZ body , which enables a functional formulation of Zhang’s projection inequality in the -order setting and yields a concise proof via covariogram–LYZ connections. The authors introduce and analyze functional radial mean bodies and Ball bodies , including new regimes for and , and study their asymptotics, linking them to and to . They establish a functional, -th order Rogers–Shephard inequality based on sup-convolution and the functional covariogram, providing sharp equality conditions tied to simplex-type structures in level sets. Overall, the paper broadens the affine-geometry toolkit for function spaces, connects Ball-type and radial-mean body concepts to the log-concave setting, and offers new structural insights with potential applications in high-dimensional convex-analytic problems and related functional inequalities.

Abstract

The Rogers-Shephard and Zhang's projection inequalities are two reverse, affine isoperimetric-type inequalities for convex bodies. Following a classical work by Schneider, both inequalities have been extended to the so-called th-order setting. In this work, we establish the th-order analogues for these inequalities in the setting of log-concave functions. Our proof of the functional Zhang's projection inequality employs properties of the asymmetric LYZ body, significantly streamlining the argument and producing a novel approach for the case . Furthermore, we introduce and analyze the radial mean bodies of a log-concave function, thereby providing a functional generalization of Gardner and Zhang's radial mean bodies. These are new even in the case . Our development leverages an extension of Ball bodies, which may be of independent interest.
Paper Structure (16 sections, 27 theorems, 206 equations)

This paper contains 16 sections, 27 theorems, 206 equations.

Table of Contents

  1. Introduction
  2. Affine isoperimetric-type inequalities and their higher-order extensions.
  3. The theory of log-concave functions.
  4. Radial mean bodies.
  5. Functional, higher-order Rogers-Shephard inequality
  6. Preliminaries
  7. The LYZ body of log-concave functions
  8. Sets associated with functions
  9. The Mellin transform of functions in one variable.
  10. Ball bodies.
  11. The $m$th-Order, Functional Setting: Zhang's Projection inequality
  12. The Polar Projection Body of a Function
  13. The Functional Covariogram
  14. The approach by radial mean bodies
  15. The Rogers-Shephard inequality in the Functional, $m$th-Order Setting
  16. ...and 1 more sections

Key Result

Theorem 1.3

Fix $n,m\in\mathbb{N}.$ Let $f\in \operatorname{LC}_n$. Then, where equality holds if and only if $f(x)=\|f\|_\infty e^{-\|x-x^\prime\|_{\Delta_n}}$ for some $x^\prime\in\mathbb{R}^n$ and an $n$-dimensional simplex $\Delta_n$ containing the origin.

Theorems & Definitions (55)

  • Definition 1.1
  • Definition 1.2
  • Theorem 1.3: The $m$th-order Zhang's projection inequality for log-concave functions
  • Proposition 1.4
  • Corollary 1.5
  • Theorem 1.6
  • Proposition 1.7
  • Theorem 1.8
  • Theorem 1.9
  • Proposition 2.1
  • ...and 45 more