Extremum properties of hexagonal partitioning and the uniform distribution in Euclidean location

TLDR: For Euclidean location problems with uniformly distributed customers, the results imply that hexagonal partitioning of the region is asymptotically optimal and that the uniform distribution is the worst possible.

Abstract: A zero-sum game with a maximizer that selects a point x in given polygon R in the plane and a minimizer that selects K points $c_1 ,c_2 , \cdots ,c_K $ in the plane is considered; the payoff is the Euclidean distance from x to the closest of the points $c_j $, or any monotonically nondecreasing function of this quantity. Lower and upper bounds on the value of the game are derived 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 induce a covering of R by K congruent regular hexagons. The analysis shows that these strategies are asymptotically optimal (for $K \to \infty $). For Euclidean location problems with uniformly distributed customers, the results imply that hexagonal partitioning of the region is asymptotically optimal and that the uniform distribution is asymptotically the worst possible.