Centreres

Centreres are a theoretical construct used in geometry and network theory to describe a central region or subset that minimizes a centrality-based potential across a space or graph. The term is not part of standard, widely adopted terminology, but it is used in some discussions to refer to the locus of points that best balance a set of reference points according to a chosen distance metric.

Definition and interpretation. Consider a metric space X and a finite multiset P of reference points p_i with positive weights w_i. Define the centrality function F(x) = sum_i w_i d(x, p_i), where d is the metric. The centrere of P is the set of points x in X that minimize F(x). If F has a unique minimizer, the centrere is a single point; if not, the centrere may be a connected set, even a segment or region in certain spaces. In graphs, the centrere of a vertex set S consists of the vertices that minimize the sum of distances to S, a discrete analogue of the same idea.

Relation to known concepts. The centrere concept generalizes familiar ideas such as the geometric median (the minimizer of a sum of distances) and the Fermat–Weber problem. Depending on the metric and the reference points, the centrere can coincide with or differ from the classical center, centroid, or other centrality notions used in statistics and graph theory.

Examples and applications. In one dimension, with two reference points at 0 and 1 and equal weights, every point in the interval [0, 1] minimizes the sum of distances, illustrating non-uniqueness. Applications of centrere ideas include facility location, clustering, and network analysis, where a central region represents an optimal area for resource placement or influence.

See also. Center, centroid, geometric median, Fermat point, Fermat–Weber problem.