Convex subdivisions with low stabbing numbers
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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 phrasesstabbing number convex subdivision space partition extremal bound
Mathematics subject classification numbers05B45 05D99 52C22
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