arXiv research

Study on semiconcavity of solutions to gradient obstacle problems on compact manifolds.

Suppose $f(x,y) + \fracκ{2} \|x\|^2 - \fracσ{2}\|y\|^2$ is convex where $σ>0$, and the argmin function $γ(x) = \{ γ: \inf_y f(x,y) = f(x,γ)\}$ exists and is single valued. We will prove $γ$ is differentiable almost everywhere. As an application we deduce a minimum principle for certain semiconcave subsolutions.

The following is a compilation of some techniques in Alexandrov's geometry which are directly connected to convexity.

Constructs retractions of CAT(1) spaces to convex subsets.

New stability bounds for Sinkhorn's algorithm in entropic optimal transport.

The paper analyzes stability and convergence rates of entropic and Sinkhorn potentials.

We discuss various characterizations of synthetic upper Ricci bounds for metric measure spaces in terms of heat flow, entropy and optimal transport. In particular, we present a characterization in terms of semiconcavity of the entropy along certain Wasserstein geodesics which is stable under convergence of mm-spaces. A…

Establishes a Lorentzian Lasry-Lions regularization theorem for functions on globally hyperbolic spacetimes.