We present two new algorithms for finding the minimum-measure simplex determined by a set of n points in R^d for arbitrary d >/= 2. The first algorithm runs in time O(n^d log n) using O(n) space. The only data structure used by this algorithms a stack. The second algorithm runs in time O(n^d) using O(n^2) space, which matches the best known time bounds for this problem in all dimensions and exceeds the previous best space bounds for all d > 3. We also present a new optimal algorithm for building one-dimensional multiplicatively weighted Voronoi diagrams that runs in linear time if the points are already sorted.

We introduce 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, which were based on high order Voronoi diagrams. Our technique allows us for the first time to efficiently dynamize our algorithms, to generalize them to higher dimensions, to find minimal convex k-vertex polygons and polytopes, and to improve many previous results. We achieve many of our results via a new algorithm for finding rectilinear nearest neighbors in the plane in time O(n log n +kn). Finally, we demonstrate a related method for finding k-point sets with minimum boundary measure or volume in arbitrary dimensions, generalizing our results for minimizing perimeter and an earlier result of the first author for minimizing area.