A lower bound for Moser's worm problem with rotations and translations
Tejasvi Singh Tomar · Zenodo (CERN European Organization for Nuclear Research) · 2026
Moser's worm problem asks for the least area of a convex set in the plane that contains a copy of every arc of length 1. We consider the version in which the copies may be rotated and translated but not reflected, and give a computer-assisted proof that the least area M+ satisfies M+ ≥ 0.2405. The previous lower bound, 0.239, was proved by Keller for the version with reflections and carries over to this one. Like the earlier lower bounds, ours uses that such a set contains the convex hull of copies of finitely many arcs. We take five polygonal arcs: a segment, an equilateral two-link arc, a three-link arc and two zigzag arcs that turn in opposite directions. Without reflections, a placed zigzag can serve only as a copy of the zigzags that turn in its own direction. On a box of placements we bound the hull area from below by signed areas of polygons spanned by vertices of the placed arcs. We bound these areas by second-order Taylor expansions with certified eigenvalue bounds and, where the hull changes its combinatorial type, by convex combinations of such bounds. Our branch and bound over a twelve-dimensional box of placements produced a certificate with 203,209,418 leaves, which a separate checker verifies in ball arithmetic. The certificate and the programs are archived at doi:10.5281/zenodo.23543309. The programs are also available at github.com/tejasvi/moser-worm-rotations-bound.