From Blaschke--Santal\'o-type inequalities to uniform contractions

TL;DR

This paper proves new inequalities for r-ball bodies, extends results on ball intersections under contractions, and confirms Alexander's conjecture for uniform contractions in Euclidean space.

Contribution

It introduces Blaschke--Santaló-type inequalities for r-ball bodies and extends the Kneser--Poulsen conjecture results, leading to a proof of Alexander's conjecture for uniform contractions.

Findings

Established Blaschke--Santaló-type inequalities for r-ball bodies.

Extended results on intersections of balls under uniform contractions.

Provided a proof of Alexander's conjecture for uniform contractions.

Abstract

In this short note, we establish Blaschke--Santal\'o-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.