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$L_p$ functional Busemann-Petty centroid inequality

Julian Haddad, Carlos Hugo Jimenez, Leticia Alves da Silva

TL;DR

The paper develops a functional analogue of the $L_p$ Busemann-Petty centroid inequality by introducing a functional mixed-volume framework for pairs of functions \(f,g\) with $1\le r< n$, and proving sharp lower bounds for integral expressions involving gradients and $p$-th powers. It establishes both the general case \(1<r<n\) and the endpoint case \(r=1\), deriving explicit sharp constants and extremizers of the form \(f(x)=bF_r(\|A x\|_2)\) and \(g(x)=aG_{p,\lambda}(\|A x\|_2)\) up to isotropic linear changes of variables. The results rely on a blend of level-set decompositions, the $L_r$- and $L_p$-Brunn–Minkowski theory, and extensions to compact domains, thereby linking Sobolev-type inequalities for functions with centroid/moment body inequalities in convex geometry. Overall, the work broadens the functional-analytic toolkit for affine geometric inequalities and clarifies the role of equality cases as ellipsoidal profiles.

Abstract

If $K\subset\mathbb{R}^n$ is a convex body and $Γ_pK$ is the $p$-centroid body of $K$, the $L_p$ Busemann-Petty centroid inequality states that $\vol(Γ_pK) \geq \vol(K)$, with equality if and only if $K$ is an ellipsoid centered at the origin. In this work, we prove inequalities for a type of functional $r$-mixed volume for $1 \leq r < n$, and establish as a consequence, a functional version of the $L_p$ Busemann-Petty centroid inequality. \keywords{Convex body, Moment body, Busemann-Petty centroid} }

$L_p$ functional Busemann-Petty centroid inequality

TL;DR

The paper develops a functional analogue of the Busemann-Petty centroid inequality by introducing a functional mixed-volume framework for pairs of functions with , and proving sharp lower bounds for integral expressions involving gradients and -th powers. It establishes both the general case and the endpoint case , deriving explicit sharp constants and extremizers of the form \(f(x)=bF_r(\|A x\|_2)\) and \(g(x)=aG_{p,\lambda}(\|A x\|_2)\) up to isotropic linear changes of variables. The results rely on a blend of level-set decompositions, the - and -Brunn–Minkowski theory, and extensions to compact domains, thereby linking Sobolev-type inequalities for functions with centroid/moment body inequalities in convex geometry. Overall, the work broadens the functional-analytic toolkit for affine geometric inequalities and clarifies the role of equality cases as ellipsoidal profiles.

Abstract

If is a convex body and is the -centroid body of , the Busemann-Petty centroid inequality states that , with equality if and only if is an ellipsoid centered at the origin. In this work, we prove inequalities for a type of functional -mixed volume for , and establish as a consequence, a functional version of the Busemann-Petty centroid inequality. \keywords{Convex body, Moment body, Busemann-Petty centroid} }

Paper Structure

This paper contains 6 sections, 9 theorems, 78 equations.

Table of Contents

  1. Introduction
  2. Some notations and tools from Convex Geometry
  3. Preliminary results
  4. Proof of the main results
  5. Case $1 < r < n$:
  6. Proof of Theorem \ref{['mainthm']}: Case $r=1$

Key Result

Theorem 1.1

Let $f$ be a $C^1$ function and $g$ a continuous non-negative function, both with compact support in $\mathbb{R}^n$, then for $1 \leq r < n$, $q = \frac{nr}{n-r}$ and $\lambda \in \left(\frac{n}{n+p}, 1\right)\cup (1, \infty)$, The sharp constant $C_{n,p, \lambda}$ is computed in Section $3$ and equality is attained if and only if $f$ and $g$ have the following forms for positive constants $a, b

Theorems & Definitions (17)

  • Theorem 1.1
  • Lemma 2.1: Lemma 4.1 of LYZoptsobnorms
  • Definition 2.2
  • Theorem 2.3
  • Definition 2.4
  • Theorem 2.5
  • Lemma 3.1: Lemma 3.2 of Zaffsob
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • ...and 7 more