A generalization of Jung's theorem
In 1901 \textit{H. W. E. Jung} had established a relation between the circumradius and the diameter of a convex body [J. Reine Angew. Math. 123, 241-257 (1901)]. The diameter of a convex body can be interpreted as the maximum of the circumradii of all 1-dimensional sections or 1- dimensional orthogonal projections of that body. Therefore the author suggests two kinds of i-dimensional circumradii of a \(d\)-dimensional convex body (\(1\leq i\leq d\)), using the section or the projection view. For these circumradii of a convex body he establishes two inequalities between circumradii of dimensions \(i\) und \(j\) (\(1\leq j\leq i\leq d\)) with equality exactly for a regular simplex. His inequalities can be interpreted as a generalization of the classical theorem of Jung, namely for \(i=j=1\) both inequalities give Jung's theorem. To prove these theorems it suffices to consider simplices since the circumradius of a convex body is determined by the circumradius of a certain simplex belonging to that body. In a similar way one can define series of inradii. Some of these functionals are studied by \textit{P. Gritzmann} and \textit{V. Klee} [Discrete Comput. Geom. 7, No. 3, 255-280 (1992)].
scientific article; zbMATH DE number 57586