Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2607.24483

The exact solution of Bellman's lost-in-a-forest problem for the golden gnomon

Alexander Temerev·Alessio Doria
View original source ↗Read source document ↗Export metadata ↓

Abstract

We solve Bellman's lost-in-a-forest problem for the golden gnomon $G$, the isosceles triangle with equal sides $1$ and apex angle $108^\circ$: the shortest curve guaranteed to reach the boundary of $G$ from an unknown starting position and heading is a symmetric seven-piece path of segments, circular shoulders, and tangents, of exactly determined length $C=1.282676025459\ldots$. To our knowledge, this is the first proved exact optimum for an isosceles triangle whose base angle is below $45^\circ$. The curve's parameters come from one isolated quartic root, and $C$ is transcendental. Equivalently, $C^{-1}G$ is the smallest homothetic golden-gnomon cover of all unit arcs. The proof introduces a balanced support calibration: one weighted family of escape inequalities, built on the linear relation among the triangle's three normals, exactly saturated by the candidate, through eighteen exact support windows, and confronting every shorter competitor at once. Aggregation along the normal fan compresses the calibration to a finite zero-sum family of supported vectors; summation by parts then bounds its total by path length whenever the running suffix balance, the ledger, stays in the unit disk. A local two-gap surgery and cyclic bitonicity force a shortest hypothetical counterexample into exactly the temporal order the ledger tolerates. Lean 4 verifies the two finite algebraic certificate families and the reusable discrete ledger identities and bounds.

Explore related subjects

math.MG

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Alexander Temerev, Alessio Doria. 2026-07-27. The exact solution of Bellman's lost-in-a-forest problem for the golden gnomon. https://arxiv.org/abs/2607.24483

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

$β$-Uniform Convexity and Divisible Domains

Divisible convex sets have long been important in the study of Hilbert geometries. When a divisible convex set is an ellipsoid, the Hilbert geometry it induces is the hyperbolic space. In general, strictly convex divisible domains exhibit negative curvature properties, but only the ellipsoid is a CAT(0) space. The notion of p-uniform convexity from the theory of Banach spaces has been proposed by Shin-Ichi Ohta as a generalization of the Alexandrov-Toponogov comparison theorems to Finsler manifolds. We prove that a natural Finsler metric on a strictly convex divisible domain is $β$-uniformly convex, where the constant $β$ is related to the regularity of the boundary. We use this to show, with AI assistance, that the Hilbert metric, under suitable local and scale-dependent assumptions, is $β$-uniformly convex on such domains.

math.MG↗
arXiv2026

A positive solution to the $L^p$ projection centroid conjecture

In a classical paper [21] in 2000, Lutwak-Yang-Zhang established the $L^p$ analog of the Petty projection inequality and the $L^p$ analog of the Busemann-Petty centroid inequality. In Section 7 of [21], Lutwak-Yang-Zhang proposed the important $L^p$ projection centroid conjecture. We give a positive solution to the $L^p$ projection centroid conjecture in this work.

math.MG↗
arXiv2026

Minimal central slices of the regular simplex

We prove that minimal-volume hyperplane sections of the regular simplex through its centroid are parallel to a facet. The proof combines variational methods with Fourier-analytic techniques and zero-diminishing arguments to show that every critical normal vector has at most three distinct non-zero coordinates. Analysis of the two- and three-value cases then yields the sharp lower bound.

math.MG↗