A lower bound for Moser's worm problem with rotations and translations

In its convex form, 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. For the version in which the copies may be rotated and translated but not reflected, we prove that this least area M⁺ satisfies M⁺ ≥ 0.2414; the best bound before, 0.239, follows from Keller's bound for the version with reflections. The proof bounds the area of the convex hull of copies of five polygonal arcs, among them two zigzags that turn in opposite directions, so that without reflections the hull must contain a zigzag turning each way. A branch and bound over the placements of the arcs, with second-order Taylor bounds for signed areas of polygons, produces a certificate, which is checked in ball arithmetic and in the Lean proof assistant.

This version revises the presentation; the results are unchanged from version 7. The certificate, the programs, the Lean files and the data of the two formal proofs are archived at doi:10.5281/zenodo.23717801. The programs and the Lean files are also available at github.com/tejasvi/moser-worm-rotations-bound/tree/v8, commit 83604e287a33dc81d6aaaa104f209896a4e8ac90.