Shakhar Smorodinsky

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 ⊆ S of size [d+3/2] such that any ball containing A contains at least roughly 4/5ed ³n 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.