On 1-regular and 1-uniform metric measure spaces
David Bate
TL;DR
This paper resolves the 1-dimensional Besicovitch problem and classifies 1-uniform metric measure spaces by introducing a tangent-space framework for 1-regular spaces. It proves a sharp decomposition of 1-regular spaces into a 1-rectifiable part and a purely 1-unrectifiable part, guided by the dichotomy of tangent spaces being either $\mathbb{R}$ or $\lambda\mathcal{S}$, and it classifies all 1-uniform spaces as isometric to a scaled ${\mathbb R}$, ${\mathcal S}$, or ${\mathcal T}$. The work then leverages these insights to obtain Besicovitch-type results in concrete classes, including doubling geodesic spaces, uniformly convex Banach spaces, and spaces with dilations such as the Heisenberg group, demonstrating rectifiability whenever $\mathcal{S}$ cannot embed. Together, these results provide a robust, structural understanding of rectifiability in metric spaces via tangents and Besicovitch configurations, with broad implications for geometric measure theory in non-Euclidean settings.
Abstract
A metric measure space $(X,μ)$ is 1-regular if \[0< \lim_{r\to 0} \frac{μ(B(x,r))}{r}<\infty\] for $μ$-a.e $x\in X$. We give a complete geometric characterisation of the rectifiable and purely unrectifiable part of a 1-regular measure in terms of its tangent spaces. A special instance of a 1-regular metric measure space is a 1-uniform space $(Y,ν)$, which satisfies $ν(B(y,r))=r$ for all $y\in Y$ and $r>0$. We prove that there are exactly three 1-uniform metric measure spaces.