Extremal covers (Q2639343)

A set \(K\subset {\mathbb{R}}^ n\) is called a cover if every set in \({\mathbb{R}}^ n\) of diameter 1 is contained in the image of K under some rigid motion of \({\mathbb{R}}^ n.\) The classical theorem of Jung asserts that the ball of diameter \(\sqrt{2n/(n+1)}\) is a cover. The author shows that the average width of any cover is at least \(\sqrt{2n/(n+1)}\). He also shows the existence of a bounded cover of least volume (which turns out not to be the ball), extending the 2-dimensional result of \textit{M. D. Kovalev} [Math. Notes 40, 736-739 (1986); translation from Mat. Zametki 40, No.3, 401-406 (1986; Zbl 0638.52004)].