Improved bounds for universal convex covers of unit arcs
Abstract
Moser's worm problem asks for a planar region of least area containing a congruent copy of every unit arc. We show that the infimum area α among convex universal covers satisfies 0.239≤α≤0.24633…, reducing the gap between the previous refereed bounds by over 75%. For the lower bound, we choose four unit polygonal arcs and prove by finite subdivision that, however they are placed, their convex hull has area at least $0.239$. For the upper bound, we construct a quadrilateral of area 0.24633… and prove cover universality by showing that its support inequalities force uncovered arcs to have length greater than one. The full proof is formalized in Lean 4 and verified by the Lean kernel. Code and certificates are available at https://github.com/ethan-keller/moser-worm-improved-bounds.
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BibTeX
@misc{keller2026improved,
title = {{Improved bounds for universal convex covers of unit arcs}},
author = {Ethan Keller},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.21968},
eprint = {2609.21968},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.21968},
}
RIS
TY - UNPB
TI - Improved bounds for universal convex covers of unit arcs
AU - Ethan Keller
PY - 2026
DA - 2026/09/18
PB - arXiv
DO - 10.48550/arxiv.2609.21968
UR - https://arxiv.org/abs/2609.21968
AB - Moser's worm problem asks for a planar region of least area containing a congruent copy of every unit arc. We show that the infimum area α among convex universal covers satisfies 0.239≤α≤0.24633…, reducing the gap between the previous refereed bounds by over 75%. For the lower bound, we choose four unit polygonal arcs and prove by finite subdivision that, however they are placed, their convex hull has area at least $0.239$. For the upper bound, we construct a quadrilateral of area 0.24633… and prove cover universality by showing that its support inequalities force uncovered arcs to have length greater than one. The full proof is formalized in Lean 4 and verified by the Lean kernel. Code and certificates are available at https://github.com/ethan-keller/moser-worm-improved-bounds.
ER -