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.

Source: arXiv:2601.21867v1 - http://arxiv.org/abs/2601.21867v1 PDF: https://arxiv.org/pdf/2601.21867v1 Original Link: http://arxiv.org/abs/2601.21867v1