Bi-Lipschitz embeddings of $SRA$-free spaces into Euclidean spaces

Vladimir Zolotov

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arXiv · 1906.02477

Bi-Lipschitz embeddings of $SRA$-free spaces into Euclidean spaces

Vladimir Zolotov

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Abstract

$SRA$-free spaces is a wide class of metric spaces including finite dimensional Alexandrov spaces of non-negative curvature, complete Berwald spaces of nonnegative flag curvature, Cayley Graphs of virtually abelian groups and doubling metric spaces of non-positive Busemann curvature with extendable geodesics. This class also includes arbitrary big balls in complete, locally compact $CAT(k)$-spaces $(k \in \mathbb R)$ with locally extendable geodesics, finite-dimensional Alexandrov spaces of curvature $\ge k$ with $k \in R$ and complete Finsler manifolds satisfying the doubling condition. We show that $SRA$-free spaces allow bi-Lipschitz embeddings in Euclidean spaces. As a corollary we obtain a quantitative bi-Lipschitz embedding theorem for balls in finite dimensional Alexandrov spaces of curvature bounded from below conjectured by S. Eriksson-Bique. The main tool of the proof is an extension theorem for bi-Lipschitz maps into Euclidean spaces. This extension theorem is close in nature with the embedding theorem of J. Seo and may be of independent interest.

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BibTeX RIS

Vladimir Zolotov. 2019-06-06. Bi-Lipschitz embeddings of $SRA$-free spaces into Euclidean spaces. https://arxiv.org/abs/1906.02477

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