Weintroduce a new method for finding several types of optimal k-point sets, minimizing perimeter, diameter, circumradius, and related measures, by testing sets of the O(k) nearest neighbors to each point. We argue that this is better in a number of ways than previous algorithms, whichwere based on high order Voronoi diagrams. Our technique allows us for the first time to efficiently maintain minimal sets as new points are inserted, to generalize our algorithms to higher dimensions, to find minimal convex k-vertex polygons and polytopes, and to improvemany previous results. Weachievemany of our results via a new algorithm for finding rectilinear nearest neighbors in the plane in time O(n log n+kn). We also demonstrate a related technique for finding minimum area k-point sets in the plane, based on testing sets of nearest vertical neighbors to each line segment determined by a pair of points. A generalization of this technique also allows us to find minimum volume and boundary measure sets in arbitrary dimensions.

Given a convex polygon P with m vertices and a set S of n points in the plane, we consider the problem of finding a placement of P that contains the maximum number of points in S. We allow both translation and rotation. Our algorithm is self-contained and utilizes the geometric properties of the containing regions in the parameter space of transformations. The algorithm requires O(nk 2 m 2 log(mk)) time and O(n +m) space, where k is the maximum number of points contained. This provides a linear improvement over the best previously known algorithm when k is large (\Theta(n)) and a cubic improvement when k is small. We also show that the algorithm can be extended to solve bichromatic and general weighted variants of the problem. 1 Introduction A planar rigid motion ae is an affine transformation of the plane that preserves distance (and therefore angles and area also). We say that a polygon P contains a set S of points if every point in S lies on P or in the interior of P . In th...