Relative Arbitrage: Sharp Time Horizons and Motion by Curvature

We characterize the minimal time horizon over which any equity market with $d \geq 2$ stocks and sufficient intrinsic volatility admits relative arbitrage with respect to the market portfolio. If $d\in \{2,3\}$, the minimal time horizon can be computed explicitly, its value being zero if $d=2$ and $\sqrt{3}\mathrm{/}(2\pi )$ if $d=3$. If $d\ge 4$, the minimal time horizon can be characterized via the arrival time function of a geometric flow of the unit simplex in ${\mathbb{R}}^d$ that we call the minimum curvature flow.