Promotion and Extension of General Escape-Covering Duality Relationship
Abstract
Bellman's lost-in-a-forest problem, Moser's worm problem, Lebesgue's universal covering problems, and the classical 'fits and covers' questions can be organized by a common obstruction principle: an object fits a region if some admissible group action places it in the region, whereas an escape object is one for which every placement fails to remain in the interior. In this paper, we develop a general theory of escape-covering duality for any dimension and explore several applications. For convex translation covers of all closed planar curves of Euclidean length one, we prove $ W \ge f(\sqrt3/9)+c_1>0.15545785378555,c_1=0.000009944944227372... $ Here f(√3/9) is Wetzel's classical lower bound. For a rational polytopal Minkowski gauge and prescribed rational facet directions, we express universal closed-loop cover feasibility as a rational polyhedron in the facet offsets. After removing translations, minimum volume is attained at a vertex. Refining the facet directions gives explicit two-sided rational bounds for the unrestricted fixed-gauge worm constant. Four exact benchmarks recover known sharp values. The construction uses an elementary circuit--circumradius form of the established non-fitting-polygon and capacity theory. We supply its support and ledger derivation, the finite complete test family in arbitrary dimension, a cover vertex bound, and the resulting algebraicity statement. The surrounding escape--covering framework includes endpoint placement, attainment, gauge stability, and finite-to-continuum error bounds. For nonsymmetric gauges the objects are parametrized closed curves, so orientation-dependent length is retained. Through the published capacity identity, the improved Wetzel estimate gives a product volume--capacity factor below $1.023781939711102$ for every planar ellipsoidal gauge.
Related
- Prescribed projections for a general class of maps and efficient covering by variable plane curves
- Mordukhovich Derivatives and Covering Constant for Set-Valued Metric Projection in General Banach Spaces
- A duality-preserving extension of the Worley-Sagan insertion and Haiman's mixed insertion for the hyperoctahedral group
- Symplectic singularities in Relative Langlands duality for products of general linear groups
- Space-Time Duality of Diffusion via Subordination: General Theory and Relativistic Models
BibTeX
@misc{deng2026promotion,
title = {{Promotion and Extension of General Escape-Covering Duality Relationship}},
author = {Zhipeng Deng},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.00569},
eprint = {2611.00569},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.00569},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - Promotion and Extension of General Escape-Covering Duality Relationship
AU - Zhipeng Deng
PY - 2026
DA - 2026/11/01
PB - arXiv
DO - 10.48550/arxiv.2611.00569
UR - https://arxiv.org/abs/2611.00569
AB - Bellman's lost-in-a-forest problem, Moser's worm problem, Lebesgue's universal covering problems, and the classical 'fits and covers' questions can be organized by a common obstruction principle: an object fits a region if some admissible group action places it in the region, whereas an escape object is one for which every placement fails to remain in the interior. In this paper, we develop a general theory of escape-covering duality for any dimension and explore several applications. For convex translation covers of all closed planar curves of Euclidean length one, we prove $ W \ge f(\sqrt3/9)+c_1>0.15545785378555,c_1=0.000009944944227372... $ Here f(√3/9) is Wetzel's classical lower bound. For a rational polytopal Minkowski gauge and prescribed rational facet directions, we express universal closed-loop cover feasibility as a rational polyhedron in the facet offsets. After removing translations, minimum volume is attained at a vertex. Refining the facet directions gives explicit two-sided rational bounds for the unrestricted fixed-gauge worm constant. Four exact benchmarks recover known sharp values. The construction uses an elementary circuit--circumradius form of the established non-fitting-polygon and capacity theory. We supply its support and ledger derivation, the finite complete test family in arbitrary dimension, a cover vertex bound, and the resulting algebraicity statement. The surrounding escape--covering framework includes endpoint placement, attainment, gauge stability, and finite-to-continuum error bounds. For nonsymmetric gauges the objects are parametrized closed curves, so orientation-dependent length is retained. Through the published capacity identity, the improved Wetzel estimate gives a product volume--capacity factor below $1.023781939711102$ for every planar ellipsoidal gauge.
ER -