The Lebesgue Universal Covering Problem

TL;DR

This paper improves the known minimal area of a convex universal covering for sets of diameter 1, reducing the previous best by a significant margin through refined geometric analysis.

Contribution

It presents a new, smaller universal covering with an area less than 0.8441153, refining previous results and correcting earlier claims about the area removed.

Findings

New universal covering with area < 0.8441153

Corrects previous area removal estimate of Hansen

Reduces the minimal area by approximately 2.2e-5

Abstract

In 1914 Lebesgue defined a "universal covering" to be a convex subset of the plane that contains an isometric copy of any subset of diameter 1. His challenge of finding a universal covering with the least possible area has been addressed by various mathematicians: Pal, Sprague and Hansen have each created a smaller universal covering by removing regions from those known before. However, Hansen's last reduction was microsopic: he claimed to remove an area of $6 \cdot 10^{-18}$, but we show that he actually removed an area of just $8 \cdot 10^{-21}$. In the following, with the help of Greg Egan, we find a new, smaller universal covering with area less than $0.8441153$. This reduces the area of the previous best universal covering by a whopping $2.2 \cdot 10^{-5}$.