Straight-line optimality in Bellman's lost-in-a-forest problem for Euclidean balls
Fuente:
arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2026
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| Acceso en línea: | |
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| _version_ | 1866917421965639680 |
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| author | Treeby, David Wang, Edward |
| author_facet | Treeby, David Wang, Edward |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_21867 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Straight-line optimality in Bellman's lost-in-a-forest problem for Euclidean balls Treeby, David Wang, Edward Probability Optimization and Control 60D05 (Primary) 52A40, 52A22 (Secondary) 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. |
| title | Straight-line optimality in Bellman's lost-in-a-forest problem for Euclidean balls |
| topic | Probability Optimization and Control 60D05 (Primary) 52A40, 52A22 (Secondary) |
| url | https://arxiv.org/abs/2601.21867 |