In this talk, we examine an important inequality between the L^p and L^infty transportation distances. Specifically, we address the following question: When is a positive function of the L^infty transportation distance bounded above by the L^p transportation distance? This inquiry is motivated by the fact that the L^infty transportation distance induces a natural topology in the space of probability measures, which is utilized in the analysis of minimizers of interaction energy functionals and the convergence of empirical measures.
We will review significant contributions in this domain, beginning with the work of Bouchitté, Jimenez, and Rajesh. They established an upper bound on the displacement map T - Id using c-cyclical monotonicity, under the condition that the source measure has a density bounded below by a positive number. However, their study left unresolved whether this result holds when the exponent p equals 1 for domains with dimension 2 or higher.
This open question was subsequently addressed by Jylhä and Rajala, who extended the analysis to transport plans. They characterized the set of source measures for which the Bouchitté-Jimenez-Rajesh result is valid as those with compact and connected support.
Throughout this presentation, we will analyze these results in detail and discuss their implications for the study of transportation distances.