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⋅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 a whopping 2.2⋅10−5.Comment: 11 pages, 5 jpeg figures, numerical errors correcte