Abstract
We study the disk containment problem introduced by Neumann-Lara and Urrutia and its generalization to higher dimensions. We relate the problem to centerpoints and lower centerpoints of point sets. Moreover, we show that for any set of n points in 
, there is a subset A ⊆ S of size \(\lfloor \frac{d+3}{2}\rfloor\) such that any ball containing A contains at least roughly \(\frac{4}{5ed^3}n\) points of S. This improves previous bounds for which the constant was exponentially small in d. We also consider a generalization of the planar disk containment problem to families of pseudodisks.
This research was started at the Gremo Workshop on Open Problems 2007 organized by Emo Welzl in Illgau, Switzerland, 2-6 July.
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Smorodinsky, S., Sulovský, M., Wagner, U. (2008). On Center Regions and Balls Containing Many Points. In: Hu, X., Wang, J. (eds) Computing and Combinatorics. COCOON 2008. Lecture Notes in Computer Science, vol 5092. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-69733-6_36
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DOI: https://doi.org/10.1007/978-3-540-69733-6_36
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