Straight-line optimality in Bellman's lost-in-a-forest problem for Euclidean balls

TL;DR

This paper proves that straight-line paths minimize expected escape time in a Euclidean ball for the Bellman lost-in-a-forest problem, using geometric conjectures and calculating expected distances.

Contribution

It establishes the optimality of straight-line paths for a min-mean variant of Bellman's problem in Euclidean balls, extending geometric methods to higher dimensions.

Findings

Straight lines minimize expected escape time among all unit-speed paths.

The minimal expected escape time is explicitly calculated for Euclidean balls.

The proof combines the Kneser--Poulsen conjecture with polygonal chain straightening results.

Abstract

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.