Metric spaces with small rough angles and the rectifiability of rough self-contracting curves

Abstract

The small rough angle ($\mbox{SRA}$) condition, introduced by Zolotov in arXiv:1804.00234, captures the idea that all angles formed by triples of points in a metric space are small. In the first part of the paper, we develop the theory of metric spaces $(X,d)$ satisfying the $\mbox{SRA}(\alpha)$ condition for some $\alpha<1$. Given a metric space $(X,d)$ and $0<\alpha<1$, the space $(X,d^\alpha)$ satisfies the $\mbox{SRA}(2^\alpha-1)$ condition. We prove a quantitative converse up to bi-Lipschitz change of the metric. We also consider metric spaces which are $\mbox{SRA}(\alpha)$ free (there exists a uniform upper bound on the cardinality of any $\mbox{SRA}(\alpha)$ subset) or $\mbox{SRA}(\alpha)$ full (there exists an infinite $\mbox{SRA}(\alpha)$ subset). Examples of SRA free spaces include Euclidean spaces, finite-dimensional Alexandrov spaces of non-negative curvature, and Cayley graphs of virtually abelian groups; examples of $\mbox{SRA}$ full spaces include the sub-Riemannian Heisenberg group, Laakso graphs, and Hilbert space. We study the existence or nonexistence of $\mbox{SRA}(\epsilon)$ subsets for $0<\epsilon<2^\alpha-1$ in metric spaces $(X,d^\alpha)$ for $0<\alpha<1$. In the second part of the paper, we apply the theory of metric spaces with small rough angles to study the rectifiability of roughly self-contracting curves. In the Euclidean setting, this question was studied by Daniilidis, Deville, and the first author using direct geometric methods. We show that in any $\mbox{SRA}(\alpha)$ free metric space $(X,d)$, there exists $\lambda_0 = \lambda_0(\alpha)>0$ so that any bounded roughly $\lambda$-self-contracting curve in $X$, $\lambda \le \lambda_0$, is rectifiable. The proof is a generalization and extension of an argument due to Zolotov, who treated the case $\lambda=0$, i.e., the rectifiability of self-contracting curves in $\mbox{SRA}$ free spaces.