By K?roly Bezdek

ISBN-10: 1441905995

ISBN-13: 9781441905994

ISBN-10: 1441906002

ISBN-13: 9781441906007

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Extra resources for Classical topics in discrete geometry

Example text

HS ln . Finally, let Il (K) be the smallest number of l-dimensional open great-hemispheres of Sd−1 that illuminate K. Obviously, I0 (K) ≥ I1 (K) ≥ · · · ≥ Id−2 (K) ≥ Id−1 (K) = 2. Let L ⊂ Ed be a linear subspace of dimension l, 0 ≤ l ≤ d−1 in Ed . The lth order circumscribed cylinder of K generated by L is the union of translates of L that have a nonempty intersection with K. Then let Cl (K) be the smallest number of translates of the interiors of some lth order circumscribed cylinders of K the union of which contains K.

Pn illuminates K if each boundary point of K is illuminated by at least one of the point sources p1 , p2 , . . , pn . Finally, the smallest n for which there exist n exterior points of K that illuminate K is called the illumination number of K denoted by I(K). In 1960, Hadwiger [155] raised the following amazingly elementary, but very fundamental question. An equivalent but somewhat different-looking concept of illumination was introduced by Boltyanski in [78]. , unit vectors) instead of point sources for the illumination of convex bodies.

In order to state their theorem in a proper form we need to introduce the following notion. If we are given a covering of a space by a system of sets, the star number of the covering is the supremum, over sets of the system, of the cardinals of the numbers of sets of the system meeting a set of the system. On the one hand, the standard Lebesgue “brick-laying” construction provides an example, for each positive integer d, of a lattice covering of Ed by closed cubes with star number 2d+1 − 1. On the other hand, Theorem 1 of [127] states that the star number of a lattice covering of Ed by translates 26 3 Coverings by Homothetic Bodies - Illumination and Related Topics of a centrally symmetric convex body is always at least 2d+1 − 1.