The greatest distance between any two points of a set, equivalently the supremum of pairwise distances in a metric space; for a circle or sphere the diameter equals twice the radius.

Principle

Diameter measures global extent: it is the supremal pairwise distance and thus an overall scale parameter that bounds every pairwise distance within the set.

Demonstration

For a circle of radius r the diameter is 2r; a bounded subset of Euclidean space has finite diameter, while an unbounded set has infinite diameter; discrete point clouds have diameter given by the farthest pair.

Misapplication

Assuming the diameter is always realized by a pair of points (the sup may not be a max in non-compact metric spaces) or confusing diameter with chord length in arbitrary nonconvex domains.

Consequence

Diameter provides an immediate bound for distances inside the set, controls notions of boundedness and compactness, and appears in inequalities (e.g., Lipschitz constants, covering numbers).

Reversal

Replacing global maximal distance by a local radial measure yields radius or inradius concepts; using average pairwise distance gives a central-tendency contrast to maximal spread.

Boundary

Defined in any metric space but may be infinite; in non-metric contexts an analogue must be specified. Diameter of graphs, manifolds, and sets must be interpreted with the appropriate distance function.

Semantic Tension

Tension with related concepts like width, circumdiameter, or maximal chord length arises: in convex bodies diameter equals longest chord, but in general sets these notions diverge.

Synthesis

Diameter is the supremum of distances between points of a set — the single-number measure of the set's maximal spread that determines boundedness and global scale.