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The longest segment in the complement of a packing

Part of: Discrete geometry

Published online by Cambridge University Press:  26 February 2010

K. Böröczky Jr  and
G. Tardos
K. Böröczky Jr
Affiliation:
Rényi Institute of Mathematics, Budapest, P.O. Box 127, 1364 Hungary
G. Tardos
Affiliation:
Rényi Institute of Mathematics, Budapest, P.O. Box 127, 1364 Hungary
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Extract

Let K be a compact convex body in ℝn not contained in a hyperplane, and denote the norm whose unit ball is ½(K − K) by ║·║k. Given a translative packing of K, we are interested in how long a segment (with respect to ║·║K) can lie in the complement of the interiors of the translates. The main result of this note is to show the existence of a translative packing such that the length of the longest segments avoiding it is only exponential in the dimension n (see below). We start here with a lower bound, showing that this bound is close to optimal for balls.

MSC classification

Secondary: 52C17: Packing and covering in $n$ dimensions
Type
Research Article
Information
Mathematika , Volume 49 , Issue 1-2 , June 2002 , pp. 45 - 49
Copyright
Copyright © University College London 2002

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