We study the following question: What is the largest deterministicamount of time T∗ that a suitably normalized martingale X can be keptinside 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) asit undergoes a geometric flow that we call (positive) minimum curvatureflow. This result has close links to the literature on stochastic andgame representations of geometric flows. Moreover, the minimum curvatureflow can be viewed as an arrival time version of the Ambrosio–Sonercodimension-(d − 1) mean curvature flow of the 1-skeleton of K. Wepresent very preliminary sampling-based numerical approximations to thesolution of the corresponding PDE. The numerical part is work in progress.This work is based on a collaboration with Camilo Garcia Trillos, MartinLarsson, and Yufei Zhang.