diametern

Diametern, commonly called the diameter in English, is a concept in geometry and metric spaces that describes the greatest distance between any two points in a given set. Formally, in a metric space (X,d) the diameter of a nonempty subset S ⊆ X is diam(S) = sup{ d(x,y) : x,y ∈ S }. If S is bounded, diam(S) is finite; if S is compact, the supremum is attained by some pair x,y ∈ S.

In Euclidean space, common special cases include: for a circle of radius r, the diameter is 2r; for a line segment of length L, the diameter is L; for a sphere of radius r, the diameter is 2r.

For a polygon or any finite point set, the diameter is the maximum distance between two of its points; in a triangle this equals the longest side.

Properties: The diameter is invariant under translation and scales linearly with uniform scaling: diam(aS) = |a| diam(S). It provides a simple measure of the size or extent of a set and complements other size measures such as radius or circumdiameter.

Applications: Used in computational geometry, clustering, and shape analysis to quantify the overall extent of a dataset or geometric object.

Origin: The term derives from Greek dia through metron, meaning “measured through.”