An Upper Bound for Lebesgue's Covering Problem

TL;DR

This paper refines existing methods to improve the upper bound on the smallest convex shape covering all sets of unit diameter in the plane, achieving a new upper bound of approximately 0.8441.

Contribution

It introduces refined techniques to tighten the upper bound for Lebesgue's covering problem, surpassing previous bounds.

Findings

New upper bound of 0.8440935944 for the covering area

Refined methods extend previous constructions

Improved bounds contribute to solving Lebesgue's problem

Abstract

A covering problem posed by Henri Lebesgue in 1914 seeks to find the convex shape of smallest area that contains a subset congruent to any point set of unit diameter in the Euclidean plane. Methods used previously to construct such a covering can be refined and extended to provide an improved upper bound for the optimal area. An upper bound of 0.8440935944 is found.