The Lebesgue universal covering problem

• Main Authors: John C. Baez, Karine Bagdasaryan, Philip Gibbs
• Format: Article
• Language: English
• Published: Carleton University 2015-09-01
• Series: Journal of Computational Geometry
• Online Access: http://jocg.org/index.php/jocg/article/view/198

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: Pál, 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 $2.2 \cdot 10^{-5}$.