geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...

, 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 center of the circle and whose endpoints lie on the circle. It can also be defined as the longest chord of the circle. Both definitions are also valid f ...

of a set of points in any

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

and the

radius In classical geometry, a radius (plural, : radii) 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 name comes from the latin ''radius'', ...

of the minimum enclosing ball of that set. It is named after

Heinrich Jung Heinrich Wilhelm Ewald Jung (4 May 1876, Essen – 12 March 1953, Halle (Saale)) was a German mathematician, who specialized in geometry and algebraic geometry. Biography Heinrich Jung was born as the son of a ''Bergrat'' (a mining officer of high ...

, 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 mathematical problem of computing the smallest circle that contains all of ...

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 by making precise the idea of a space having no "punctures" or "missing endpoints", 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 a line segment between the two points. It can be calculated from the Cartesian coordinates of the points using the Pythagorean theorem, therefore ...

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 defi ...

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 points in a plane that are at a given distance from a given point, the centre. Equivalently, it is the curve traced out by a point that moves in a plane so that its distance from a given point is cons ...

enclosing all points whose radius satisfies :

r \leq \frac,

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

equilateral triangle In geometry, an equilateral triangle is a triangle in which all three sides have the same length. In the familiar Euclidean geometry, an equilateral triangle is also equiangular; that is, all three internal angles are also congruent to each oth ...

(or its three vertices) one has

r = \frac.

General metric spaces

For any

bounded set :''"Bounded" and "boundary" are distinct concepts; for the latter see boundary (topology). A circle in isolation is a boundaryless bounded set, while the half plane is unbounded yet has a boundary. In mathematical analysis and related areas of m ...

S

in any

metric space In mathematics, a metric space is a set together with a notion of '' distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general setti ...

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 degenerate triangles, but ...

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 such as the

Manhattan distance A taxicab geometry or a Manhattan geometry is a geometry whose usual distance function or metric of Euclidean geometry is replaced by a new metric in which the distance between two points is the sum of the absolute differences of their Cartesian co ...

in the plane,

r=d/2

: any two closed balls of radius

d/2

centered at points of

S

have a non-

empty Empty may refer to: ‍ Music Albums * ''Empty'' (God Lives Underwater album) or the title song, 1995 * ''Empty'' (Nils Frahm album), 2020 * ''Empty'' (Tait album) or the title song, 2001 Songs * "Empty" (The Click Five song), 2007 * ...

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, thei ...

, 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 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