Sparse convex hull coverage

Given a set P of n data points and an integer k, a fundamental computational task is to find a smaller subset Q⊆P of only k points which approximately preserves the geometry of P. Here we consider the problem of finding the subset Q of k points which best captures the convex hull of P, where our error measure is the sum of the distances of the points in P to the convex hull of Q. We generalize the problem to allow the set R that we must select Q from to differ from P, as well as to allow more general functions of the distances of the uncovered points of P, such as other norms or weighted distance functions.