##  [Diameter](/diameter-0) 

 Definition

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

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

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

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

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

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

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

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

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.