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Covering the Plane with Congruent Copies of a Convex Body
Author(s) -
Kuperberg W.
Publication year - 1989
Publication title -
bulletin of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 2.396
H-Index - 48
eISSN - 1469-2120
pISSN - 0024-6093
DOI - 10.1112/blms/21.1.82
Subject(s) - mathematics , convex body , regular polygon , plane (geometry) , combinatorics , geometry , lattice (music) , point (geometry) , convex hull , mathematical analysis , physics , acoustics
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 .