Escaping the unit ball

TL;DR

This work tackles the min-mean version of Bellman’s Lost in a Forest for forests shaped as unit balls. It proves that straight-line paths minimize the expected escape time $J(\gamma)$ by representing this expectation as $J(\gamma)=\frac{1}{\pi}\int_{0}^{\infty} \mathrm{area}(S_{\gamma}(t))\,dt$ and applying the Kneser--Poulsen conjecture in the plane, together with polygonal-chain straightening arguments; the authors extend the result to $\mathbf{R}^n$ using higher-dimensional expansion results. They then derive exact closed-form expressions for the minimal escape time of straight-line paths, showing $J(\ell)=\frac{2}{\sqrt{\pi}}\,\frac{\Gamma\left(\frac{n+2}{2}\right)}{\Gamma\left(\frac{n+3}{2}\right)}$, with the 2D value $J(\ell)=\frac{8}{3\pi}$. The findings connect geometric probability with convex-geometry inequalities, providing precise escape-time constants for ball-shaped forests in any dimension.

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.

Figures (2)

• Figure 1: A straight line path is an expansion of any non-linear path.
• Figure 2: A linear escape path for the unit disk $D$.

Theorems & Definitions (8)

• Theorem 2.1
• Theorem 2.2: Kneser--Poulsen in the plane
• proof : Proof of Theorem \ref{['thm:2d']}
• Theorem 3.1: Bezdek--Connelly
• Theorem 3.2
• proof
• Proposition 4.1
• proof