Escaping the unit ball

We prove that among all unit-speed paths, a straight line minimises the expected escape time from a ball in $\mathbf{R}^n$, solving the min-mean variant of Bellman’s Lost in a Forest problem for ball-shaped forests. The proof uses the Kneser–Poulsen conjecture in the plane, together with results on polygonal chain straightening in higher dimensions. Moreover, we calculate this minimal escape time by deriving the expected linear distance to the boundary of a ball in $n$ dimensions.

💡 Research Summary

The paper addresses the “Lost in a Forest” problem introduced by Bellman, focusing on the min‑mean (expected‑time) variant rather than the more commonly studied min‑max version. The setting is a unit ball (or disk) in ℝⁿ, with the hiker’s starting point uniformly distributed inside the ball and the hiker moving at unit speed along a prescribed path γ. The goal is to determine which path minimizes the expected escape time
 J(γ)=E