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From Blaschke--Santaló-type inequalities to uniform contractions

Károly Bezdek

TL;DR

The paper establishes Blaschke--Santaló-type inequalities for $r$-ball bodies and leverages them to extend uniform-contraction results related to the Kneser--Poulsen conjecture in Euclidean space. By combining Brunn--Minkowski inequalities for intrinsic volumes with Alexandrov-type comparisons, it shows ${\rm V}_{k}(A^{r})\le{\rm V}_{k}(B^{r})$ with equality precisely when the $r$-convex hull of $A$ is a ball, and derives a sharp volume-product bound ${P}_{k}(\mathbf{A})\le P_{k}(\mathbf{B}^d[\mathbf{o}, r/2])$, maximized at radius $r/2$. A strengthened KP-type result for uniform contractions is proved for sufficiently large $N$, using the BS-type inequality and Jung’s estimate, which in particular yields Alexander’s conjecture in the plane for all uniform contractions. These results deepen the understanding of ball-polyhedral geometries via intrinsic volumes and highlight extremality by balls in the $r$-ball setting.

Abstract

In this short note, we establish Blaschke--Santaló-type inequalities for $r$-ball bodies. Building on these inequalities, we somewhat further extend earlier results on analogues of the Kneser--Poulsen conjecture concerning intersections of balls under uniform contractions in Euclidean $d$-space. As an immediate corollary, we obtain a proof of Alexander's conjecture for uniform contractions.

From Blaschke--Santaló-type inequalities to uniform contractions

TL;DR

The paper establishes Blaschke--Santaló-type inequalities for -ball bodies and leverages them to extend uniform-contraction results related to the Kneser--Poulsen conjecture in Euclidean space. By combining Brunn--Minkowski inequalities for intrinsic volumes with Alexandrov-type comparisons, it shows with equality precisely when the -convex hull of is a ball, and derives a sharp volume-product bound , maximized at radius . A strengthened KP-type result for uniform contractions is proved for sufficiently large , using the BS-type inequality and Jung’s estimate, which in particular yields Alexander’s conjecture in the plane for all uniform contractions. These results deepen the understanding of ball-polyhedral geometries via intrinsic volumes and highlight extremality by balls in the -ball setting.

Abstract

In this short note, we establish Blaschke--Santaló-type inequalities for -ball bodies. Building on these inequalities, we somewhat further extend earlier results on analogues of the Kneser--Poulsen conjecture concerning intersections of balls under uniform contractions in Euclidean -space. As an immediate corollary, we obtain a proof of Alexander's conjecture for uniform contractions.
Paper Structure (4 sections, 6 theorems, 21 equations)

This paper contains 4 sections, 6 theorems, 21 equations.

Table of Contents

  1. Introduction
  2. Proof of Theorem \ref{['BS-type']}
  3. Proof of Corollary \ref{['volume product inequality']}
  4. Proof of Theorem \ref{['KP-type improved']}

Key Result

Theorem 1

Let $r>0$. Let $d\geq 2$ and $1\leq k\leq l\leq d$ be integers. If $\emptyset\neq A\subset \mathbb{E}^d$ is a compact set with $0<{\rm cr}(A)\leq r$ and $B:=\mathbf{B}^d[\mathbf{o}, R^{\mathbb{E}^d}_{r, l}(A)]$, then with equality if and only if ${\rm conv}_r(A)$ is a ball of radius $R^{\mathbb{E}^d}_{r, l}(A)$.

Theorems & Definitions (8)

  • Definition 1
  • Theorem 1
  • Remark 2
  • Corollary 3
  • Theorem 4
  • Corollary 5
  • Lemma 6
  • Corollary 7