Hexagonal economic regions solve the location problem. (English) Zbl 1026.90059

Morgan, Frank; Bolton, Roger

Am. Math. Mon. 109, No. 2, 165-172 (2002).

From the introduction: One practical measure of the efficiency of the distribution of centers of production is the average distance \(\rho\) from, say, uniformly distributed consumers to the nearest center, for transportation costs are roughly proportional to distance. As the number of points increases and boundary effects become negligible, the points seem to tend toward the centers of a tiling by regular hexagons. Our Theorems prove in a certain mathematical sense that regular hexagons beat any other collection of congruent or noncongruent shapes of equal or unequal areas, in finite or infinite domains. (It is clear that each region should be the “Voronoi cell” of points closest to the given center, hence generally a polygon).

MSC:

90B85	Continuous location
05B45	Combinatorial aspects of tessellation and tiling problems
51M20	Polyhedra and polytopes; regular figures, division of spaces
52C20	Tilings in \(2\) dimensions (aspects of discrete geometry)
91D10	Models of societies, social and urban evolution