Lebesgue's Universal Covering Problem

Find the minimum-area convex set in the plane that contains a congruent copy of every planar set of diameter at most 1.

Pál showed a regular hexagon of area 2√3 ≈ 3.464 works (1920); Sprague immediately clipped corners (1936) reaching ≈ 2.849; Hansen's 1992 micro-clipping stood for thirty years. In 2024 new computer-assisted corner-cutting lowered the record again — while the best lower bound (Meir–Moser's 2.219…) trails far behind. The gap between what we can cover with and what we can prove necessary is the whole story.