articles: Relaxing the nearest centre assumption in central place theory

Although a variety of modifications of classical central place theory has been proposed, one area that remains unexplored is the effect of relaxing the nearest centre assumption for the purchase of a specified basket of goods within a given hierarchical level. This article examines the effect of such a relaxation on central place market areas by using higher-order Voronoi diagrams. When used to model market areas, higher-order Voronoi diagrams can be interpreted as overlapping and probabilistic regions. These diagrams construct market areas based on the assumption that consumers choose from a set of $k (k = 1, 2, \ldots, n)$ nearest centres of the same hierarchical level. If consumers are assumed to be indifferent between the k centres, the appropriate market areas are given by the order-k Voronoi diagram. In this case, it is shown that sales potentials are consistent with those that result when the nearest centre assumption is in effect. If consumers are assumed to have a preference for nearer centres, market areas are defined by the ordered, order-k Voronoi diagram. This situation generates sales potentials which can vary between centres.