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On a Blaschke-Santaló-type inequality for $r$-ball bodies

Károly Bezdek

TL;DR

This work presents a concise proof of a Blaschke–Santaló-type inequality for intrinsic volumes of $r$-ball bodies in Euclidean space: among all compact sets of a fixed $d$-volume, the $k$-th intrinsic volume of the $r$-ball body $X^r$ is maximized by a Euclidean ball, with equality only when the original set is congruent to that ball. The proof leverages the $r$-ball convex hull, the identity $X^r=({\rm conv}_r X)^r$, and a Brunn–Minkowski framework for intrinsic volumes together with the isoperimetric inequality to establish the bound and its uniqueness. A 2D Mahler-type remark is also derived, showing that among $r$-disk domains with fixed area, the $r$-lens extremizes a related quantity, and yielding a corresponding comparison between $V_2$ of $r$-ball bodies. The results unify and extend prior Euclidean, spherical, and hyperbolic analogues and have potential implications for problems such as the Kneser–Poulsen conjecture via contractions of ball intersections.

Abstract

Let ${\mathbb E}^d$ denote the $d$-dimensional Euclidean space. The $r$-ball body generated by a given set in ${\mathbb E}^d$ is the intersection of balls of radius $r$ centered at the points of the given set. The author [Discrete Optimization 44/1 (2022), Paper No. 100539] proved the following Blaschke-Santaló-type inequality for $r$-ball bodies: for all $0<k< d$ and for any set of given $d$-dimensional volume in ${\mathbb E}^d$ the $k$-th intrinsic volume of the $r$-ball body generated by the set becomes maximal if the set is a ball. In this note we give a new proof showing also the uniqueness of the maximizer. Some applications and related questions are mentioned as well.

On a Blaschke-Santaló-type inequality for $r$-ball bodies

TL;DR

This work presents a concise proof of a Blaschke–Santaló-type inequality for intrinsic volumes of -ball bodies in Euclidean space: among all compact sets of a fixed -volume, the -th intrinsic volume of the -ball body is maximized by a Euclidean ball, with equality only when the original set is congruent to that ball. The proof leverages the -ball convex hull, the identity , and a Brunn–Minkowski framework for intrinsic volumes together with the isoperimetric inequality to establish the bound and its uniqueness. A 2D Mahler-type remark is also derived, showing that among -disk domains with fixed area, the -lens extremizes a related quantity, and yielding a corresponding comparison between of -ball bodies. The results unify and extend prior Euclidean, spherical, and hyperbolic analogues and have potential implications for problems such as the Kneser–Poulsen conjecture via contractions of ball intersections.

Abstract

Let denote the -dimensional Euclidean space. The -ball body generated by a given set in is the intersection of balls of radius centered at the points of the given set. The author [Discrete Optimization 44/1 (2022), Paper No. 100539] proved the following Blaschke-Santaló-type inequality for -ball bodies: for all and for any set of given -dimensional volume in the -th intrinsic volume of the -ball body generated by the set becomes maximal if the set is a ball. In this note we give a new proof showing also the uniqueness of the maximizer. Some applications and related questions are mentioned as well.
Paper Structure (3 sections, 4 theorems, 12 equations)

This paper contains 3 sections, 4 theorems, 12 equations.

Table of Contents

  1. Introduction
  2. Proof of Theorem \ref{['Bezdek-inequality-extended']}
  3. Proof of Remark \ref{['2D-Mahler-type']}

Key Result

Theorem 1

Let $\mathbf{A}\subset\mathbb{E}^d$, $d>1$ be a compact set of volume $V_{d}(\mathbf{A})>0$ and $r>0$. If $\mathbf{B}\subset\mathbb{E}^d$ is a ball with $V_{d}(\mathbf{A})=V_{d}(\mathbf{B})$, then holds for all $1\leq k\leq d$ with equality if and only if $\mathbf{A}$ is congruent (i.e., isometric) to $\mathbf{B}$.

Theorems & Definitions (8)

  • Definition 1
  • Theorem 1
  • Corollary 2
  • Remark 4
  • Definition 2
  • Remark 5
  • Lemma 6
  • Lemma 7