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 $2\pi /\sqrt{27}$.