Covering the Plane with Congruent Copies of a Convex Body

It is shown that every plane compact convex set K with an interiorpoint admits a covering of the plane with density smaller thanor equal to 8(23 – 3)/3 = 1.2376043.... For comparison,the thinnest covering of the plane with congruent circles isof density 2 / 27 = 1.209199576.... (see R. Kershner [3]), whichshows that the covering density bound obtained here is closeto the best possible. It is conjectured that the best possibleis 2 / 27. The coverings produced here are of the double-latticekind consisting of translates of K and translates of —K.