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We study the following question: What is the largest deterministic amount oftime $T_*$ that a suitably normalized martingale $X$ can be kept inside aconvex body $K$ in $\mathbb{R}^d$? 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 curvatureflow. This result has close links to the literature on stochastic and gamerepresentations of geometric flows. Moreover, the minimum curvature flow can beviewed as an arrival time version of the Ambrosio--Soner codimension-$(d-1)$mean curvature flow of the $1$-skeleton of $K$. Our results are obtained by amix of probabilistic and analytic methods.

Minimum curvature flow and martingale exit times