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The Upper-Bound Theorem for Families of Boxes in ℝd

Part of: General convexity

Published online by Cambridge University Press:  21 December 2009

Jürgen Eckhoff
Jürgen Eckhoff
Affiliation:
Department of Mathematics, University College London, Gower Street, London, WC1E 6BT, England.
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Abstract

Let be a family of axis-aligned parallelotopes, or boxes, in ℝd. Denote by fk () the number of subfamilies of of size k + 1 with non-empty intersection. In an earlier paper, the author proved that, if f0 () = n and fr() = 0, then fk() ≤ fk(n, d, r) for k = 1,…,r − 1, where fk(n, d, r) is some explicitly given number. The result is best possible for all k. Here it is shown that, if equality is attained for some such k, then equality is attained for each such k.

MSC classification

Secondary: 52A37: Other problems of combinatorial convexity
Type
Research Article
Information
Mathematika , Volume 54 , Issue 1-2 , December 2007 , pp. 25 - 34
Copyright
Copyright © University College London 2007

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