Coverings: Variations on a result of Rogers and on the Epsilon-net theorem of Haussler and Welzl |
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Authors: | Nóra Frankl János Nagy Márton Naszódi |
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Institution: | 1. Department of Mathematics, London School of Economics and Political Science, United Kingdom;2. Department of Geometry, Lorand Eötvös University, Pázmány Péter Sétány 1/C Budapest, 1117, Hungary |
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Abstract: | We consider four problems. Rogers proved that for any convex body , we can cover by translates of of density very roughly . First, we extend this result by showing that, if we are given a family of positive homothets of of infinite total volume, then we can find appropriate translation vectors for each given homothet to cover with the same (or, in certain cases, smaller) density.Second, we extend Rogers’ result to multiple coverings of space by translates of a convex body: we give a non-trivial upper bound on the density of the most economical covering where each point is covered by at least a certain number of translates.Third, we show that for any sufficiently large , the sphere can be covered by strips of width , where no point is covered too many times.Finally, we give another proof of the previous result based on a combinatorial observation: an extension of the Epsilon-net Theorem of Haussler and Welzl. We show that for a hypergraph of bounded Vapnik–Chervonenkis dimension, in which each edge is of a certain measure, there is a not-too large transversal set which does not intersect any edge too many times. |
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Keywords: | Covering Rogers’ bound Spherical strip Density Set-cover Epsilon-net theorem |
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