In mathematics, thediameter of a set of points in ametric space is the largest distance between points in the set. As an important special case, thediameter of a metric space is the largest distance between any two points in the space. This generalizes thediameter of a circle, the largest distance between two points on the circle. This usage of diameter also occurs in medical terminology concerning alesion or ingeology concerning a rock.

Abounded set is a set whose diameter is finite. Within a bounded set, all distances are at most the diameter.

The diameter of an object is theleast upper bound (denoted "sup") of the set of all distances between pairs of points in the object.Explicitly, if${\displaystyle S}$ is a set of points with distances measured by ametric${\displaystyle \rho }$, the diameter is^[1][2]$${\displaystyle \mathrm{diam}(S)=\underset{x,y\in S}{sup}\rho (x,y).}$$

The diameter of the empty set is a matter of convention. It can be defined to be zero,^[2][3]${\displaystyle -\mathrm{\infty }}$,^[3] or undefined.

For any bounded set in theEuclidean plane orEuclidean space, the diameter of the object or set is the same as the diameter of itsconvex hull. For anyconvex shape in theplane, the diameter is the largest distance that can be formed between two oppositeparallel linestangent to its boundary.^[4]

The diameter of a circle is exactly twice its radius. However, this is true only for a circle, and only in theEuclidean metric.Jung's theorem provides more general inequalities relating the diameter to the radius.^[5] Theisodiametric inequality orBieberbach inequality, a relative of theisoperimetric inequality, states that, for a given diameter, the planar shape with the largest area is a disk, and the three-dimensional shape with the largest volume is a sphere.^[6][7] Thepolygons of maximum area for a given diameter and number of sides are thebiggest little polygons.^[8]

Just as the diameter of a two-dimensional convex set is the largest distance between two parallel lines tangent to and enclosing the set, thewidth is often defined to be the smallest such distance.^[4] The diameter and width are equal only for abody of constant width, for which all pairs of parallel tangent lines have the same distance. Every set of bounded diameter in the Euclidean plane is a subset of a body of constant width with the same diameter.^[9]

The diameter or width of a two-dimensional point set or polygon can be calculated efficiently usingrotating calipers.^[4] Algorithms for computing diameters in higher-dimensional Euclidean spaces have also been studied incomputational geometry; seediameter (computational geometry).

Indifferential geometry, the diameter is an important globalRiemannianinvariant. Everycompact set in aRiemannian manifold, and every compact Riemannian manifold itself, has finite diameter. For instance, theunit sphere of any dimension, viewed as a Riemannian manifold, has diameter${\displaystyle \pi }$. This differs from its diameter as a subset of Euclidean space (which would equal two) because, as a Riemannian manifold, distances are measured alonggeodesics within the manifold.^[10]

In a Riemannian manifold whoseRicci curvature has a positive constant lower bound, the diameter is also bounded byMyers's theorem. According toCheng's maximal diameter theorem, the unique manifold with the largest diameter for a given curvature lower bound is a sphere with that curvature. The theorem is named afterShiu-Yuen Cheng, who published it in 1975.^[10][11]

Ingraph theory, the diameter of a connected undirected graph is the farthest distance between any two of its vertices. That is, it is the diameter of a set, for the set of vertices of the graph, and for theshortest-path distance in the graph. Diameter may be considered either for weighted or for unweighted graphs. Researchers have studied the problem of computing the diameter, both in arbitrary graphs and in special classes of graphs.

Special cases of graph diameter includethe diameter of a group, defined using aCayley graph with the largest diameter possible for a given group, and thediameter of the flip graph of triangulations of a point set, the minimum number of local moves needed to transform one triangulation into another for two triangulations chosen to be as far apart as possible.

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