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

TL;DR

This work develops the theory of metric spaces with the small rough angle condition $\mathrm{SRA}(\alpha)$ and ties it to snowflaking, proving that, up to bi-Lipschitz changes, $p$-snowflake metrics correspond to $\mathrm{SRA}(\alpha)$ spaces with $\alpha=2^{-1/p}$ via a sharp bound. It also demonstrates that snowflaked subsets can host large $\mathrm{SRA}(\varepsilon)$ configurations, including Cantor-type sets of positive dimension, and provides a rich catalog of $\mathrm{SRA}$-free vs $\mathrm{SRA}$-full spaces, ranging from Euclidean spaces to Heisenberg groups and Laakso graphs. The second part extends Zolotov’s rectifiability results to roughly self-contracting curves in $\mathrm{SRA}(\alpha)$-free spaces, showing bounded rough $\lambda$-self-monotone curves are rectifiable for small enough $\lambda$, with a Ramsey-type discrete-analytic proof. Together, these results illuminate the geometric-analytic landscape of metric spaces governed by small-angle constraints and offer tools for embedding, rectifiability, and curve theory in synthetic metric settings.

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 $\mbox{SRA}(α)$ condition for some $α<1$. Given a metric space $(X,d)$ and $0<α<1$, the space $(X,d^α)$ satisfies the $\mbox{SRA}(2^α-1)$ condition. We prove a quantitative converse up to bi-Lipschitz change of the metric. We also consider metric spaces which are $\mbox{SRA}(α)$ free (there exists a uniform upper bound on the cardinality of any $\mbox{SRA}(α)$ subset) or $\mbox{SRA}(α)$ full (there exists an infinite $\mbox{SRA}(α)$ 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 $\mbox{SRA}(ε)$ subsets for $0<ε<2^α-1$ in metric spaces $(X,d^α)$ for $0<α<1$. 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 $\mbox{SRA}(α)$ free metric space $(X,d)$, there exists $λ_0 = λ_0(α)>0$ so that any bounded roughly $λ$-self-contracting curve in $X$, $λ\le λ_0$, is rectifiable. The proof is a generalization and extension of an argument due to Zolotov, who treated the case $λ=0$, i.e., the rectifiability of self-contracting curves in $\mbox{SRA}$ free spaces.

Figures (4)

• Figure 1: Metric tree $T_t$
• Figure 2: Metric graphs $G_0$, $G_1$ and $G_2$
• Figure 3: The curve $\{(x,y):\sqrt{x^2+y^2} = 1 + \tfrac{1}{2} \sqrt{(x-1)^2+y^2}\} \cup [-3,0]\times\{0\}$
• Figure 4: Distribution of elements in $P^t$ and $P^{t+1}$

Theorems & Definitions (93)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Definition 2.1: Small rough angles condition
• Remark 2.2
• Remark 2.3
• Remark 2.4
• Remark 2.5
• Remark 2.6
• Proposition 2.7: Erdös--Füredi