A note on the diameter of small sub-Riemannian balls

TL;DR

This paper investigates the diameter of small sub-Riemannian balls in manifolds with varying regularity, showing it equals twice the radius in smooth cases and approaches this in less regular cases, regardless of bracket-generating conditions.

Contribution

It establishes the diameter of small sub-Riemannian balls in $C^{1,1}$ and $C^0$ manifolds, extending understanding beyond the bracket-generating assumption.

Findings

Diameter equals twice the radius in $C^{1,1}$ manifolds.

Diameter approaches twice the radius in $C^0$ manifolds.

Results hold independently of the bracket-generating condition.

Abstract

We observe that the diameter of small (in a locally uniform sense) balls in $C^{1,1}$ sub-Riemannian manifolds equals twice the radius. We also prove that, when the regularity of the structure is further lowered to $C^0$, the diameter is arbitrarily close to twice the radius. Both results hold independently of the bracket-generating condition.