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On the Covering Densities of Quarter-Convex Disks.pdf

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On the Covering Densities of Quarter-Convex Disks

1 23 Discrete Computational Geometry ISSN 0179-5376 Discrete Comput Geom DOI 10.1007/s00454-015-9696-8 On the Covering Densities of Quarter- Convex Disks Kirati Sriamorn Fei Xue 1 23 Your article is protected by copyright and all rights are held exclusively by Springer Science +Business Media New York. This e-offprint is for personal use only and shall not be self- archived in electronic repositories. If you wish to self-archive your article, please use the accepted manuscript version for posting on your own website. You may further deposit the accepted manuscript version in any repository, provided it is only made publicly available 12 months after official publication or later and provided acknowledgement is given to the original source of publication and a link is inserted to the published article on Springers website. The link must be accompanied by the following text: The final publication is available at ”. Discrete Comput Geom DOI 10.1007/s00454-015-9696-8 On the Covering Densities of Quarter-Convex Disks Kirati Sriamorn1 · Fei Xue1 Received: 18 November 2014 / Revised: 24 March 2015 / Accepted: 27 March 2015 ? Springer Science+Business Media New York 2015 Abstract It is conjectured that for every convex disk K , the translative covering density of K and the lattice covering density of K are identical. It is well known that this conjecture is true for every centrally symmetric convex disk. For the non- symmetric case, we only know that the conjecture is true for triangles (Januszewski in Discrete Comput Geom 43:167–178, 2010). In this paper, we prove the conjecture for a class of convex disks (quarter-convex disks), which includes all triangles and convex quadrilaterals. Keywords Translative covering · Lattice covering · Covering density · Convex disk · Quarter-convex disk Mathematics Subject Classification 05B40 · 11H31 · 52C15 1 Introduction An n-dimensional convex body is a compact convex subset of Rn with an interior point. A 2-dimensional convex bo