Locality of centred tangent cones in the Wasserstein space

TL;DR

This paper characterizes the local structure of tangent cones in the Wasserstein space by relating centered tangent fields to normal spaces of measure-supported sets, revealing a local geometric condition.

Contribution

It introduces a local characterization of centered tangent cones in Wasserstein space by linking tangent fields to normal spaces of measure-supported sets.

Findings

Centered tangent fields are concentrated on subspaces linked to measure support.

Normal spaces correspond to sets of certain dimensions where the measure is concentrated.

The characterization simplifies understanding tangent cones by local geometric conditions.

Abstract

The geometric tangent cone to a probability measure $\mu$ is a set of measure-valued applications that are almost geodesics. This is a nonlocal condition, typically lost when conditioning the measure on a given set. We show that if one removes the barycenter of any element of the tangent cone, then the resulting set of centred measure fields is characterized by a local condition. Precisely, centred tangent fields must be concentrated on a family of vector subspaces attached to any point, and these subspaces correspond to the normal spaces to some sets of ``dimension $k$'' on which the measure $\mu$ is concentrated.