Curves of constant width and Lebesgue's covering problem

Abstract

A universal cover is a convex set in the plane that contains a congruent copy of every planar set of diameter one. Lebesgue asked in 1914 for one of least area, and the value is not known. We prove that every convex universal cover has area at least 0.8344, improving on 0.832, published in 2005, and 0.833, in a 2026 preprint, both of which come from a disc together with an equilateral triangle and a regular pentagon. Our test sets are instead curves of constant width: the disc, the Reuleaux triangle and the Reuleaux pentagon. Each contains the regular polygon it is built on, so the family is strictly larger at the same number of bodies and the same number of placement parameters, and we show that the classical configuration admits an arrangement whose hull has area below 0.8336, so no bound drawn from those three sets by this argument reaches ours. Curves of constant width were proposed for this role, and explored numerically, by Gibbs in 2014; what is added here is a proof. It consists of an analytic reduction followed by one finite computation. The reduction bounds the hull area from below over an entire box of placements at once, by eroding each Reuleaux polygon to a fixed set contained in every placement that box allows. The computation is an exhaustive subdivision of the resulting five-dimensional space, recorded as a certificate of 486,799,600 nodes and checked by a verifier independent of the search, with a rigorous bound on its floating point error some thousands of times smaller than the margin the verification attains.