geometry Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...

, Jung's theorem is an inequality between the

diameter In geometry, a diameter of a circle is any straight line segment that passes through the centre of the circle and whose endpoints lie on the circle. It can also be defined as the longest Chord (geometry), chord of the circle. Both definitions a ...

of a set of points in any

Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are ''Euclidean spaces ...

and the

radius In classical geometry, a radius (: radii or radiuses) of a circle or sphere is any of the line segments from its Centre (geometry), center to its perimeter, and in more modern usage, it is also their length. The radius of a regular polygon is th ...

of the minimum enclosing ball of that set. It is named after Heinrich Jung, who first studied this inequality in 1901. Algorithms also exist to solve the

smallest-circle problem The smallest-circle problem (also known as minimum covering circle problem, bounding circle problem, least bounding circle problem, smallest enclosing circle problem) is a computational geometry problem of computing the smallest circle that cont ...

explicitly.

Statement

Consider a

compact set In mathematics, specifically general topology, compactness is a property that seeks to generalize the notion of a closed and bounded subset of Euclidean space. The idea is that a compact space has no "punctures" or "missing endpoints", i.e., i ...

K \subset \mathbb^n

and let :

d = \max_ \,  p - q \, _2

be the

of ''K'', that is, the largest

Euclidean distance In mathematics, the Euclidean distance between two points in Euclidean space is the length of the line segment between them. It can be calculated from the Cartesian coordinates of the points using the Pythagorean theorem, and therefore is o ...

between any two of its points. Jung's theorem states that there exists a

closed ball In mathematics, a ball is the solid figure bounded by a ''sphere''; it is also called a solid sphere. It may be a closed ball (including the boundary points that constitute the sphere) or an open ball (excluding them). These concepts are defin ...

with radius :

r \leq d \sqrt

that contains ''K''. The boundary case of equality is attained by the regular ''n''-

simplex In geometry, a simplex (plural: simplexes or simplices) is a generalization of the notion of a triangle or tetrahedron to arbitrary dimensions. The simplex is so-named because it represents the simplest possible polytope in any given dimension. ...

Jung's theorem in the plane

The most common case of Jung's theorem is in the plane, that is, when ''n'' = 2. In this case the theorem states that there exists a

circle A circle is a shape consisting of all point (geometry), points in a plane (mathematics), plane that are at a given distance from a given point, the Centre (geometry), centre. The distance between any point of the circle and the centre is cal ...

enclosing all points whose radius satisfies :

r \leq \frac,

and this bound is as tight as possible since when ''K'' is an

equilateral triangle An equilateral triangle is a triangle in which all three sides have the same length, and all three angles are equal. Because of these properties, the equilateral triangle is a regular polygon, occasionally known as the regular triangle. It is the ...

(or its three vertices) one has

r = \frac.

General metric spaces

For any

bounded set In mathematical analysis and related areas of mathematics, a set is called bounded if all of its points are within a certain distance of each other. Conversely, a set which is not bounded is called unbounded. The word "bounded" makes no sense in ...

S

in any

metric space In mathematics, a metric space is a Set (mathematics), set together with a notion of ''distance'' between its Element (mathematics), elements, usually called point (geometry), points. The distance is measured by a function (mathematics), functi ...

d/2\le r\le d

. The first inequality is implied by the

triangle inequality In mathematics, the triangle inequality states that for any triangle, the sum of the lengths of any two sides must be greater than or equal to the length of the remaining side. This statement permits the inclusion of Degeneracy (mathematics)#T ...

for the center of the ball and the two diametral points, and the second inequality follows since a ball of radius

d

centered at any point of

S

will contain all of

S

. Both these inequalities are tight: * In a ''uniform metric space'', that is, a space in which all distances are equal,

r=d

. * At the other end of the spectrum, in an

injective metric space In metric geometry, an injective metric space, or equivalently a hyperconvex metric space, is a metric space with certain properties generalizing those of the real line and of L∞ distances in higher- dimensional vector spaces. These properties c ...

such as the

Manhattan distance Taxicab geometry or Manhattan geometry is geometry where the familiar Euclidean distance is ignored, and the distance between two point (geometry), points is instead defined to be the sum of the absolute differences of their respective Cartesian ...

in the plane,

r=d/2

: any two closed balls of radius

d/2

centered at points of

S

have a non- empty

intersection In mathematics, the intersection of two or more objects is another object consisting of everything that is contained in all of the objects simultaneously. For example, in Euclidean geometry, when two lines in a plane are not parallel, their ...

, therefore all such balls have a common intersection, and a radius

d/2

ball centered at a point of this intersection contains all of

S

. Versions of Jung's theorem for various

non-Euclidean geometries In mathematics, non-Euclidean geometry consists of two geometries based on axioms closely related to those that specify Euclidean geometry. As Euclidean geometry lies at the intersection of metric geometry and affine geometry, non-Euclidean geo ...

are also known (see e.g. Dekster 1995, 1997).

References

* * * * * *

External links

*{{mathworld , title = Jung's Theorem , urlname = JungsTheorem Geometric inequalities Euclidean geometry Theorems in geometry Metric geometry