Jung's theorem
Jung's geometric theorem (named after Heinrich Jung ) makes a mathematical statement about how big a sphere must be in a dimensional space that encloses a given set of points.
formulation
Let there be a finite number of points and let it be the maximum Euclidean distance between two points.
Jung's theorem says that there is a -dimensional sphere with a radius such that all points lie within the sphere (including the edge).
Furthermore, the center of the sphere is clearly determined with the smallest possible radius.
Special case of a plane
Most famous is the case of points in the plane; H. . In this case Jung's theorem says that the radius is.
Exactly this radius is required for the three corner points of an equilateral triangle.
literature
- Heinrich Jung: About the smallest circle that includes a flat figure ( Memento from March 10, 2007 in the Internet Archive ), J. Reine Angew. Math. 137 (1910), 310-313
- Hans Rademacher and Otto Toeplitz : From numbers and figures (samples of mathematical thinking for lovers of mathematics) , Springer-Verlag 2000 (reprint of the 2nd edition from 1933), ISBN 3-540-63303-0 , 14th chapter