Covering the Plane with Congruent Copies of a Convex Disk
Abstract
It is shown that there exists a number
d0 < 8(2sqrt3-3)/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 d0. This improves on the previous result
by
W. Kuperberg , which showed that a density of 8(2sqrt3-3)/3 can always be obtained. Since the thinnest covering of the plane with congruent cirlcles is of density 2pi/sqrt(27)=1.20919..., we strengthen the case for the conjecture that the smallest such number d0 is 2pi/sqrt(27).