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In mathematics, the diameter of a set of points in a metric space is the largest distance between points in the set. This generalizes the diameter of a circle, the largest distance between two points on the circle. This usage of diameter also occurs in medical terminology concerning a lesion or in geology concerning a rock, where the diameter of an object is the least upper bound 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 a metric ${\displaystyle \rho }$, the diameter is $${\displaystyle \operatorname {diam} (S)=\sup _{x,y\in S}\rho (x,y).}$$

Of the empty set

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

In Euclidean spaces

For a convex shape in the plane, the diameter is the largest distance that can be formed between two opposite parallel lines tangent to its boundary. It can be calculated efficiently using rotating calipers. For a curve of constant width such as the Reuleaux triangle, the width and diameter are the same because all such pairs of parallel tangent lines have the same distance.

For any solid object or set of scattered points in ${\displaystyle n}$-dimensional Euclidean space, the diameter of the object or set is the same as the diameter of its convex hull. Algorithms for computing this kind of diameter have been studied in computational geometry; see diameter (computational geometry).

In differential geometry

In differential geometry, the diameter is an important global Riemannian invariant.

Relation to other measures

The diameter of a circle is exactly twice its radius. However, this is true only for a circle, and only in the Euclidean metric. Jung's theorem provides more general inequalities relating the diameter to the radius.

Just as the diameter of a two-dimensional convex set is the largest distance between two parallel lines tangent to and enclosing the set, the width is often defined to be the smallest such distance. Like the diameter, the width can be computed using the method of rotating calipers. The diameter and width are equal only for a body of constant width. Every set of bounded diameter in the Euclidean plane is a subset of a body of constant width with the same diameter.

References

1. Kaplansky, Irving (1977), Set Theory and Metric Spaces (2nd ed.), Chelsea Publishing, p. 69, MR 0446980
2. ^ Rado, T.; Reichelderfer, P. V. (1955), Continuous transformations in analysis. With an introduction to algebraic topology, Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen mit besonderer Berücksichtigung der Anwendungsgebiete, vol. LXXV, Springer-Verlag, p. 14, doi:10.1007/978-3-642-85989-, MR 0079620
3. ^ Capoyleas, Vasilis; Rote, Günter; Woeginger, Gerhard (1991), "Geometric clusterings", Journal of Algorithms, 12 (2): 341–356, doi:10.1016/0196-6774(91)90007-L, MR 1105480
4. ^ Toussaint, Godfried T. (1983), "Solving geometric problems with the rotating calipers" (PDF), Proc. MELECON '83: Mediterranean Electrotechnical Conference, 24–26 May 1983, Athens, IEEE, CiteSeerX 10.1.1.155.5671
5. Klee, Victor (1971), "What is a convex set?", The American Mathematical Monthly, 78: 616–631, doi:10.1080/00029890.1971.11992815, JSTOR 2316569, MR 0285985

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