The Lebesgue universal covering problem

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 · 10−18, but we show that he actually removed an area of just 8 · 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 · 10−5.