Local structure of centred tangent cones in the Wasserstein space
Averil Aussedat
TL;DR
The paper studies the geometric tangent cone in the 2-Wasserstein space for a probability measure with finite second moment, focusing on the centred part of tangent fields. It develops a local, DC$k$-based decomposition of the base measure $\\mu$ into mutually singular components $\\mu^k$ with masses $m_k$, yielding a global splitting of the centred tangent cone $\\mathbf{Tan}_{\\mu}^0$ into $\\sum_{k=0}^d m_k\\mathbf{Tan}_{\\mu^k}^0$, where each component is supported on the normal directions to a corresponding DC$k$ set $A_k$ described by a Grassmannian section $D_k$. The construction uses a decomposition into $d+1$ components and Zajíček’s DC$k$ theory to control where mass can be split along optimal transport plans, and it connects tangent/solenoidal structures to orthogonal Grassmannian descriptions $D^{\\mathbf{Sol}}$ and $D^{\\mathbf{Tan}}$. The work also relates these objects to Preiss tangent measures and contrasts them with Alberti–Marchese’s decomposability bundle, clarifying how decomposability interacts with Wasserstein tangency and metric orthogonality.
Abstract
This article investigates the geometric tangent cone to a probability measure with finite second moment. It is known that the tangent elements induced by a map belong to the $L^2_μ$ closure of smooth gradients. We show that at the opposite, the elements that have barycenter 0 are characterized by a local condition, i.e. as the barycenter-free measures that are concentrated on a family of vector subspaces attached to any point. Our results rely on a decomposition of a measure into $d+1$ components, each allowing optimal plans to split mass in a fixed number of directions. We conclude by giving some links with Preiss tangent measures and illustrating the difference with Alberti and Marchese's decomposability bundle.