Lebesgue's universal covering problem

Remarks

The problem is open: the exact minimal area is not known. Pál showed that the regular hexagon whose opposite sides are at distance $1$ is a universal cover; its area is $\sqrt{3}/2 \approx 0.866$. Cutting suitable pieces off this hexagon gives smaller universal covers, and the best known upper bound is about $0.8441$. A disc of diameter $1$ is not a universal cover: the equilateral triangle of side $1$ has diameter $1$ but circumradius $1/\sqrt{3} > 1/2$. Lower bounds come from the fact that a universal cover must contain copies of several specific sets of diameter $1$ at once. Examples are a disc of diameter $1$ and an equilateral triangle of side $1$. The best known lower bound is about $0.832$.