An extension of Jung's theorem
Jung's theorem is related to Helly's theorem and can be stated as follows: Let \(C\subset {\mathbb{R}}^ n\) be a compact convex set of unit circumradius and let B be the unit ball containing C. Then diam C is not less than \(\sqrt{(2n+2)/n}\) (which is the edge length of a regular n- simplex s inscribed in B). If diam C\(=\sqrt{(2n+2)/n}\) then C contains such a simplex s. The author's extension of Jung's theorem answers the following questions. Suppose diam C is greater than but close to \(\sqrt{(2n+2)/n}\). Then C should contain an n-simplex s' inscribed in B and close to s. How close? What is the range of edge length of s' in terms of the diameter and dimension of C? Lower bounds for the inradius and minimal width are also obtained. The proofs rely on Caratheodory theorem (If \(X\subset {\mathbb{R}}^ n\), each point of the convex hull of X is a convex combination of at most \(n+1\) points of X''), induction on the dimension and the study of some properties of simplices.
scientific article; zbMATH DE number 3917141