Extremum Properties of Hexagonal Partitioning and the Uniform Distribution in Euclidean Location

We consider a zero-sum game with a maximizer who selects a point x in given polygon R in the plane and a minimizer who selects K points Cl, c2, ..., cK in the plane; the payoff is min lix cill, or any monol<i<K tonically nondecreasing function of this quantity. We derive lower and upper bounds on the value of the game by considering, respectively, the maximizer's strategy of selecting a uniformly distributed random point in R and the minimizer's strategy of selecting K members of a (uniformly) randomly positioned grid of centers that induces a covering of R by K congruent regular hexagons. Our analysis shows that these strategies are asymptotically optimal (for K + o). For Euclidean location problems with uniformly distributed customers, our results imply that hexagonal partitioning of the region is asymptotically optimal, and that the uniform distribution is asymptotically the worst possible.