maksimietäisyys

Maksimietäisyys, in Finnish, refers to the greatest distance between two elements within a given set when a distance function or metric is defined. For a subset S of a metric space (X, d), this is often expressed as the diameter diam(S) = sup{ d(x,y) : x, y ∈ S }. If S is finite, the maximum is attained, so diam(S) = max{ d(x,y) : x, y ∈ S }.

In geometry and Euclidean spaces, distance is commonly the Euclidean distance. For a finite collection of points, the maximum distance between any pair of points equals the set’s diameter. In the plane or higher dimensions, the points that realize this maximum typically lie on the convex hull of the set, and efficient methods exist to find an antipodal pair without checking all pairs.

In graph theory, the concept is adapted to discrete structures. The distance between two vertices is the length of a shortest path connecting them. The maximum distance from a vertex v to any other vertex is its eccentricity e(v). The graph’s diameter is diam(G) = max_v e(v) = max_{u,v} d(u,v). These notions underpin analyses of network spread, navigation, and robustness.

Edge cases include infinite diameter: in unbounded spaces like the whole of R^n with the usual metric, the diameter is infinite. Computationally, finite sets allow straightforward pairwise comparison; larger or structured sets may use geometric or graph-based optimizations. Applications span clustering, facility location, and assessments of spatial or network dispersion.