New Algorithms for Minimum Area k-gons

Given a set P of n points in the plane, we wish to find a set Q ⊂ P of k points for which the convex hull conv(Q) has the minimum area. We solve this, and the related problem of finding a minimum area convex k-gon, in time O(n log n) and space O(n log n) for fixed k, almost matching known bounds for the minimum area triangle problem. Our algorithm is based on finding a certain number of nearest vertical neighbors to each line segment determined by two input points. We use a classical result of Ramsey theory to prove that these nearest neighbors suffice to determine the minimum convex k-gon.