What is the largest deterministic amount of time T∗ that a suitably normalized martingale X can be kept inside a convex body K in Rd? We show, in a viscosity framework, that T∗ equals the time it takes for the relative boundary of K to reach X(0) as it undergoes a geometric flow that we call (positive) minimum curvature flow. This result has close links to the literature on stochastic and game representations of geometric flows. Moreover, the minimum curvature flow can be viewed as an arrival time version of the Ambrosio–Soner codimension-(d − 1) mean curvature flow of the 1-skeleton of K. We present very preliminary sampling-based numerical approximations to the solution of the corresponding PDE. The numerical part is work in progress.
This work is based on a collaboration with Camilo Garcia Trillos, Martin Larsson, and Yufei Zhang.