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