Covering a Convex Body by Its Negative Homothetic Copies
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Survey Paper Combinatorial problems on the illumination of convex bodies
This is a review of various problems and results on the illumination of convex bodies in the spirit of combinatorial geometry. The topics under review are: history of the Gohberg–Markus–Hadwiger problem on the minimum number of exterior sources illuminating a convex body, including the discussion of its equivalent forms like the minimum number of homothetic copies covering the body; generalizat...
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We estimate the number and ratio of negative homothetic copies of a d-dimensional convex body C sufficient for the covering of C. If the number of those copies is not very large, then our estimates are better than recent estimates of Rogers and Zong. Particular attention is paid to the 2-dimensional case. It is proved that every planar convex body can be covered by two copies of ratio −4 3 (thi...
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Some geometric inequalities for convex bodies, where the equality cases characterize simplices, are improved in the form of stability estimates. The inequalities all deal with covering by homothetic copies. Mathematics Subject Classification (2000). Primary 52A40, Secondary 52A21.
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The covering number c(K) of a convex body K is the least number of smaller homothetic copies of K needed to cover K . We provide new upper bounds for c(K) when K is centrally symmetric by introducing and studying the generalized α -blocking number βα 2 (K) of K . It is shown that when a centrally symmetric convex body K is sufficiently close to a centrally symmetric convex body K′ , then c(K) i...
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In 1957, H. Hadwiger conjectured that a convex body K in a Euclidean d-space, d 1, can always be covered by 2 smaller homothetic copies of K. We verify this conjecture when K is the polar of a cyclic d-polytope. Mathematics Subject Classifications (1991): 52A15, 52A20.
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تاریخ انتشار 2006