[Paper Review] Sets with small angles in self-contracted curves

Vladimir Zolotov|arXiv (Cornell University)|Apr 1, 2018

Advanced Differential Geometry Research4 references3 citations

TL;DR

This paper establishes rectifiability of bounded self-contracted curves in a broad class of metric spaces by introducing the concept of 'SRA(α)-free' spaces, which are metric spaces without large subspaces with small rough angles. It proves that non-rectifiable self-contracted curves must contain arbitrarily large SRA(α) subspaces, and thus bounded self-contracted curves are rectifiable in spaces like reversible $C^inity$-Finsler manifolds, locally compact $ extrm{CAT}(k)$-spaces with extendable geodesics, and Busemann spaces under the same conditions.

ABSTRACT

We study metric spaces with bounded rough angles. E. Le Donne, T. Rajala and E. Walsberg implicitly used this notion to show that infinite snowflakes can not be isometrically embedded into finite dimensional Banach spaces. We show that bounded non-rectifiable self-contracted curves contain metric subspaces with bounded rough angles. Which provides rectifiability of bounded self-contracted curves in a wide class of metric spaces including reversible $C^{\infty}$-Finsler manifolds, locally compact $CAT(k)$-spaces with locally extendable geodesics and locally compact Busemann spaces with locally extendable geodesics. We also extend the result on non embeddability of infinite snowflakes to this class of spaces.

Motivation & Objective

• To establish rectifiability of bounded self-contracted curves in a wide class of metric spaces beyond Euclidean and Riemannian settings.
• To identify a geometric obstruction—subspaces with small rough angles (SRA(α) subspaces)—that prevents rectifiability.
• To show that the absence of such subspaces (i.e., SRA(α)-freeness) implies rectifiability of self-contracted curves.
• To extend the non-embeddability result of infinite snowflakes into finite-dimensional Banach spaces to a broader class of metric spaces.
• To unify and generalize prior results on rectifiability in $ extrm{CAT}(0)$, Riemannian, and normed spaces using a new intrinsic geometric framework.

Proposed method

• Introduces the notion of $ extrm{SRA}({eta})$ spaces, defined by the inequality $ d(x,y) \leq \max\{ d(x,z) + \alpha d(z,y), \alpha d(x,z) + d(z,y) \} $, which captures metric spaces with uniformly small rough angles.
• Defines discrete self-expanding (DSE) spaces as finite metric sets satisfying $ d(x_i,x_j) \leq d(x_i,x_k) $ for $ i \leq j \leq k $, modeling the discrete structure of self-contracted curves.
• Proves that if the total length $ L(X) $ of a DSE space exceeds $ C({\alpha},k) \cdot D(X) $, then it contains a $ k $-point $ \textrm{SRA}({\alpha}) $ subspace (Theorem 2).
• Establishes that $ \textrm{SRA}({\alpha}) $-free spaces—those without $ k $-point $ \textrm{SRA}({\alpha}) $ subspaces for some $ k $—admit rectifiable bounded self-contracted curves.
• Applies Ramsey-theoretic and pigeonhole arguments to show that spaces satisfying $ \textrm{LEG} $ and $ \textrm{LRB} $ conditions are locally $ \textrm{SRA}({\alpha}) $-free.
• Uses contradiction: if a curve is non-rectifiable, its image must contain arbitrarily large $ \textrm{SRA}({\alpha}) $ subspaces, contradicting local $ \textrm{SRA}({\alpha}) $-freeness in the target spaces.

Experimental results

Research questions

• RQ1 Can the rectifiability of bounded self-contracted curves be established in general metric spaces beyond Euclidean and Riemannian settings?
• RQ2 What intrinsic geometric property prevents rectifiability in self-contracted curves?
• RQ3 Under what conditions on a metric space do self-contracted curves remain rectifiable?
• RQ4 Can the non-embeddability of infinite snowflakes into finite-dimensional Banach spaces be extended to other classes of metric spaces?
• RQ5 How do geometric conditions like extendable geodesics and curvature bounds relate to the absence of $ \textrm{SRA}({\alpha}) $ subspaces?

Key findings

• Every non-rectifiable self-contracted curve in a metric space contains arbitrarily large $ \textrm{SRA}({\alpha}) $ subspaces for any $ \frac{1}{2} < \alpha < 1 $, as formalized in Theorem 2.
• Bounded self-contracted curves are rectifiable in complete reversible $ C^\infty $-Finsler manifolds, due to their local $ \textrm{SRA}({\alpha}) $-freeness.
• Bounded self-contracted curves are rectifiable in complete locally compact $ \textrm{CAT}(k) $-spaces with locally extendable geodesics, as these spaces satisfy the $ \textrm{LEG} $ and $ \textrm{LRB} $ conditions.
• Bounded self-contracted curves are rectifiable in complete locally compact Busemann spaces with locally extendable geodesics, again due to local $ \textrm{SRA}({\alpha}) $-freeness.
• The non-embeddability of infinite snowflakes into finite-dimensional Banach spaces extends to all $ \textrm{SRA}({\alpha}) $-free spaces, including the aforementioned classes.
• The doubling condition on balls implies that $ \textrm{SRA}({\alpha}) $-free sets in such spaces are uniformly bounded in size, which supports the rectifiability result via pigeonhole arguments.