# Self-intersection of Optimal geodesics

created by cavallett on 29 Nov 2012
modified on 25 Aug 2014

[BibTeX]

Published Paper

Inserted: 29 nov 2012
Last Updated: 25 aug 2014

Journal: Bull. London Math. Soc.
Year: 2013

Abstract:

Let $(X,d,m)$ be a geodesic metric measure space. Consider a geodesic $\mu_{t}$ in the $L^{2}$-Wasserstein space. Then as $s$ goes to $t$, the support of $\mu_{s}$ and the support of $\mu_{t}$ have to overlap, provided an upper bound on the densities holds. We give a more precise formulation of this self-intersection property. Given a geodesic of $(X,d,m)$ and $t\in[0,1]$, we consider the set of times for which this geodesic belongs to the support of $\mu_{t}$. We prove that $t$ is a point of Lebesgue density 1 for this set, in the integral sense. Our result applies to spaces satisfying $\mathsf{CD}(K,\infty)$. The non branching property is not needed.

Keywords: optimal transportation