Beitr\"age zur Algebra und Geometrie Contributions to Algebra and Geometry Volume 35 (1994), No. 1, 101-108. Covering the Plane with Congruent Copies of a Convex Disk Edwin H. Smith Abstract. It is shown that there exists a number $d_0\hskip-2pt< 8(2\sqrt{3}-3)/3=1.237604..$, such that every compact convex set $K$ with an interior point admits a covering of the plane with density smaller than or equal to $d_0$. This improves on the previous result [9], which showed that a density of $8(2\sqrt{3}-3)/3$ can always be obtained. Since the thinnest covering of the plane with congruent circles is of density $2\pi /\sqrt{27}= 1.20919\ldots$, we strengthen the case for the conjecture that the smallest such number $d_0$ is $2\pi /\sqrt{27}$.