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Periodica Mathematica Hungarica

, Volume 57, Issue 2, pp 217–225 | Cite as

Convex subdivisions with low stabbing numbers

  • Csaba D. Tóth
Article

Abstract

It is shown that for every subdivision of the d-dimensional Euclidean space, d ≥ 2, into n convex cells, there is a straight line that stabs at least Ω((log n/log log n)1/(d−1)) cells. In other words, if a convex subdivision of d-space has the property that any line stabs at most k cells, then the subdivision has at most exp(O(k d−1 log k)) cells. This bound is best possible apart from a constant factor. It was previously known only in the case d = 2.

Key words and phrases

stabbing number convex subdivision space partition extremal bound 

Mathematics subject classification numbers

05B45 05D99 52C22 

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Copyright information

© Springer Science+Business Media B.V. 2008

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsUniversity of CalgaryAlbertaCanada

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