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 R~d, there is a subset A is contained in S of size [(d+3)/2] such that any ball containing A contains at least roughly 4/(5ed3) points of 5. 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.
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