Snowflake universality of Wasserstein spaces
Alexandr Andoni, Assaf Naor, Ofer Neiman

TL;DR
This paper demonstrates the universality of Wasserstein spaces over -dimensional Euclidean space for embedding finite metric spaces with low distortion, revealing their rich geometric structure and implications for metric geometry open problems.
Contribution
It establishes sharp embedding results for Wasserstein spaces, showing their universality for finite metric spaces and implications for Alexandrov spaces and expander graphs.
Findings
Wasserstein spaces -dimensional are universal for finite metric spaces with low distortion
Sharpness of embedding exponents for Wasserstein spaces when p
Existence of metric spaces that do not embed into Wasserstein spaces with low distortion
Abstract
For let denote the metric space of all -integrable Borel probability measures on , equipped with the Wasserstein metric . We prove that for every , every and every finite metric space , the metric space embeds into with distortion at most . We show that this is sharp when in the sense that the exponent cannot be replaced by any larger number. In fact, for arbitrarily large there exists an -point metric space such that for every any embedding of the metric space into incurs distortion that is at least a constant multiple of . These statements establish that…
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Taxonomy
TopicsGeometric Analysis and Curvature Flows · Geometry and complex manifolds · Advanced Operator Algebra Research
