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Inequalities for sections and projections of log-concave functions

Natalia Tziotziou

TL;DR

This paper extends classical convex-geometry inequalities on sections and projections to integrable log-concave functions by introducing functional analogues $\Psi_k(f)$ and $\Phi_k(f)$ of the dual and affine quermassintegrals and proving sharp bounds in terms of $\|f\|_1$ and $f(0)$. It further develops a suite of results linking sections and projections through Milman-type inequalities, Radon-transform quotients, and slicing, with functional versions that carry constants of the same order as their geometric counterparts. The main technical toolkit includes Ball's $K_p(f)$ bodies, mixed-integral theory, and Kubota-type formulas, enabling translation between function-level problems and convex-body inequalities. The results also extend to densities of $s$-concave measures and yield Shephard-type projection bounds with universal constants, highlighting the robustness of log-concavity in preserving sharp inequality constants. Overall, the work broadens the functional-analytic framework for slicing and projection problems, offering new tools and sharper constants for log-concave settings.

Abstract

We provide extensions of geometric inequalities about sections and projections of convex bodies to the setting of integrable log-concave functions. Namely, we consider suitable generalizations of the affine and dual affine quermassintegrals of a log-concave function $f$ and obtain upper and lower estimates for them in terms of the integral $\|f\|_1$ of $f$, we give estimates for sections and projections of log-concave functions in the spirit of the lower dimensional Busemann-Petty and Shephard problem, and we extend to log-concave functions the affirmative answer to a variant of the Busemann-Petty and Shephard problems, proposed by V. Milman. The main goal of this article is to show that the assumption of log-concavity leads to inequalities in which the constants are of the same order as that of the constants in the original corresponding geometric inequalities.

Inequalities for sections and projections of log-concave functions

TL;DR

This paper extends classical convex-geometry inequalities on sections and projections to integrable log-concave functions by introducing functional analogues and of the dual and affine quermassintegrals and proving sharp bounds in terms of and . It further develops a suite of results linking sections and projections through Milman-type inequalities, Radon-transform quotients, and slicing, with functional versions that carry constants of the same order as their geometric counterparts. The main technical toolkit includes Ball's bodies, mixed-integral theory, and Kubota-type formulas, enabling translation between function-level problems and convex-body inequalities. The results also extend to densities of -concave measures and yield Shephard-type projection bounds with universal constants, highlighting the robustness of log-concavity in preserving sharp inequality constants. Overall, the work broadens the functional-analytic framework for slicing and projection problems, offering new tools and sharper constants for log-concave settings.

Abstract

We provide extensions of geometric inequalities about sections and projections of convex bodies to the setting of integrable log-concave functions. Namely, we consider suitable generalizations of the affine and dual affine quermassintegrals of a log-concave function and obtain upper and lower estimates for them in terms of the integral of , we give estimates for sections and projections of log-concave functions in the spirit of the lower dimensional Busemann-Petty and Shephard problem, and we extend to log-concave functions the affirmative answer to a variant of the Busemann-Petty and Shephard problems, proposed by V. Milman. The main goal of this article is to show that the assumption of log-concavity leads to inequalities in which the constants are of the same order as that of the constants in the original corresponding geometric inequalities.

Paper Structure

This paper contains 6 sections, 19 theorems, 144 equations.

Table of Contents

  1. Introduction
  2. Background information and auxiliary results
  3. Affine and dual affine quermassintegrals
  4. Sections versus projections
  5. Sections of log-concave functions
  6. Projections of log-concave functions

Key Result

Theorem 1.1

Let $f:{\mathbb R}^n\to [0,+\infty)$ be a geometric log-concave integrable function. Then, for every $1\leqslant k\leqslant n-1$, where $\psi_{n,k}:=\min\{L_n,\sqrt{n/k}\,\sqrt{\log(en/k)}\}$ and $c>0$ is an absolute constant.

Theorems & Definitions (31)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • ...and 21 more