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. ${\displaystyle n}$

formulation

Let there be a finite number of points and let it be the maximum Euclidean distance between two points. ${\displaystyle a_{1},a_{2},\dots ,a_{k}\in \mathbb {R} ^{n}}$ ${\displaystyle d:=\max \nolimits _{i,j=1,\dots ,k}\|a_{i}-a_{j}\|_{2}}$

Jung's theorem says that there is a -dimensional sphere with a radius such that all points lie within the sphere (including the edge). ${\displaystyle n}$ ${\displaystyle r\leq d{\sqrt {\tfrac {n}{2(n+1)}}}}$ ${\displaystyle a_{1},a_{2},\dots ,a_{k}}$

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. ${\displaystyle n=2}$ ${\displaystyle r\leq d/{\sqrt {3}}}$

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