Isodiametric inequality in Carnot groups

TL;DR

This paper investigates the failure of the classical isodiametric inequality in Carnot groups, showing that certain distances do not maximize volume for fixed diameter, and explores related measure and rectifiability issues.

Contribution

It demonstrates that in Carnot groups, the isodiametric inequality can fail under various distances and links this failure to properties of minimizing curves and measure comparisons.

Findings

Existence of homogeneous distances in Carnot groups where the inequality fails

Failure of the inequality for Carnot-Caratheodory distances with finite-time non-minimizing curves

Connections between measure discrepancies, rectifiability, and the generalized 1/2-Besicovitch conjecture

Abstract

The classical isodiametric inequality in the Euclidean space says that balls maximize the volume among all sets with a given diameter. We consider in this paper the case of Carnot groups. We prove that for any Carnot group equipped with a Haar measure one can find a homogeneous distance for which this fails to hold. We also consider Carnot-Caratheodory distances and prove that this also fails for these distances as soon as there are length minimizing curves that stop to be minimizing in finite time. Next we study some connections with the comparison between Hausdorff and spherical Hausdorff measures, rectifiability and the generalized 1/2-Besicovitch conjecture giving in particular some cases where this conjecture fails.