March 18, 2014 – Gershon Wolansky, Technion

Gershon Wolansky, Technion

Title: From optimal transportation to optimal teleportation via the critical Sobolev embedding

In this talk I review the notion of metric differentiability of measure valued parths in the Wasserstein p-metric(W_p) and concentrate on the case where the tangent derivtive of this paths is itself a measure. It turns out that the existence of a metric derivative is conditioned on some notion of dimensionality of the measure, and on the connectivity of its support. In case of a connected support, the existence of a metric derivative is guranteed only if p’>d, where p’=p/(p-1) and d is the dimension associated with the measure. If p'<d then the path is only q=1/d+1/p Holder, and for p’=d, corresponding to the critical Sobolev embedding, the path is log-Lipshitz in W_p. The case of disconnected component corresponds to d=\infty, and the path is 1/p-Holder. In the later case I show a sharp estimate of the Holder norm, depending only on a finite number of parameters and corresponding to a metric transport on a finite graph.