TejasviTomar
October 2026 · Preprint

A lower bound of 0.836 for Lebesgue's universal covering problem

Tejasvi Singh Tomar

In plain words

In 1914 Lebesgue asked for the least area of a convex patch that can be placed over every set of points in the plane at most 1 apart. Gibbs's patch of area 0.8441 is the smallest known. Lower bounds come from a few test shapes that a patch must hold at once. The best earlier bounds were 0.83474, by Hatchett, and 0.8344, by Mishra. Using a disk and the Reuleaux triangle, pentagon and heptagon, all of width 1, this paper raises the bound to 0.836, with a computer-checked certificate.

Abstract

A universal cover is a convex set in the plane that contains a congruent copy of every set of diameter 1. Lebesgue asked for the least area Λ of a universal cover. The best upper bound is Λ ≤ 0.8440935944, due to Gibbs. The previous lower bounds were Λ ≥ 0.83474, due to Hatchett, and Λ ≥ 0.8344, due to Mishra. We prove Λ ≥ 0.836. Like the earlier lower bounds, the proof uses the fact that a universal cover contains the convex hull of congruent copies of any finitely many sets of diameter 1. We use the disk and the regular Reuleaux triangle, pentagon and heptagon of width 1, whose relative placements form an eight-dimensional space. The main ingredient is a second-order lower bound for the hull area on a box of placements. We follow finitely many points of the bodies as they move. While these points stay in angular order around a point of the disk, the area of the polygon they span is at most the hull area. This area is a polynomial in the translations and in the sines and cosines of the angles of the placement. A Taylor expansion with a certified bound for the smallest eigenvalue of the Hessian bounds it on the whole box. A checker in ball arithmetic verifies the resulting certificate, which has 1,552,274 nodes.

Left, a two-by-two grid of four shapes: a grey disk with a dashed outline, a yellow rounded triangle, a magenta rounded pentagon and a pale green rounded heptagon. Right, the same four outlines overlap, almost concentric, inside a larger teal convex region with straight edges and curved stretches; thick black, magenta and green pieces of its boundary and coloured dots mark where each shape touches it.
Illustration of the test bodies. Left: the disk of diameter 1 and the regular Reuleaux triangle, pentagon and heptagon of width 1. Right: the four bodies in the placement that the paper's local minimization finds, with their convex hull shaded teal; its area is 0.83649…. The hull touches every body: the disk along the black arcs, the pentagon along the magenta arcs, the heptagon along a short green arc, and corners of the three Reuleaux polygons at the dots; straight segments join these pieces. A universal cover is convex and contains copies of all four bodies, so it contains the hull of some placement; the paper proves that every placement gives a hull of area at least 0.836.