  We study the following question: What is the largest deterministic amount of
time $T_*$ that a suitably normalized martingale $X$ can be kept inside a
convex 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 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 $K$. Our results are obtained by a
mix of probabilistic and analytic methods.
