A note on flatness of non separable tangent cone

TL;DR

This paper investigates the geometric structure of tangent cones in Alexandrov spaces with curvature bounds, showing that the support of certain measures lies within a Hilbert space even without separability assumptions.

Contribution

It establishes that the support of the pushforward measure on the tangent cone is contained in a Hilbert space without requiring the cone to be separable.

Findings

Support of the pushforward measure is in a Hilbert space

Results hold without tangent cone separability

Extends understanding of tangent cone geometry in Alexandrov spaces

Abstract

Given a probability measure P on an Alexandrov space S with curvature bounded below, we prove that the support of the pushforward of P on the tangent cone at its (exponential) barycenter is a subset of a Hilbert space, without separability of the tangent cone.