Enclosing k points in the smallest axis parallel rectangle

We consider the following clustering problem. Given a set S of n points in the plane, and given an integer k, n/2 < k ≤ n, we want to find the smallest axis parallel rectangle (smallest perimeter or area) that encloses exactly k points of S. We present an algorithm which runs in time O(n + k(n - k)²) improving previous algorithms which run in time O(k²n) and do not perform well for larger k values. We present an algorithm to enclose k of n given points in an axis parallel box in d-dimensional space which runs in time O(dn + dk(n - k)^2(d-1)) and occupies O(dn) space. We slightly improve algorithms for other problems whose runtimes depend on k.