Covering the Plane with Congruent Copies of a Convex Body

It is shown that every plane compact convex set K with an interior point admits a covering of the plane with density smaller than or equal to 8(2√3 − 3)/3 = 1.2376043…. For comparison, the thinnest covering of the plane with congruent circles is of density 2π / √27 = 1.209199576…. (see R. Kershner [ 3 ]), which shows that the covering density bound obtained here is close to the best possible. It is conjectured that the best possible is 2π / √27. The coverings produced here are of the double‐lattice kind consisting of translates of K and translates of — K .