The minimum mean width translation cover for sets of diameter one (Q1267346)

Let \(K\) be a convex body in \(E^d\) and \(w_K(u)\) be its width in the direction \(u\in S^{d-1}\). Denote by \(w(K)\) the mean width of \(K\), i.e. put \[ w(K)= {1\over w_d} \int_{S^{d-1}} w_K(u)dS \] where \(w_d=2 \pi^{p/2} \Gamma ({d\over 2})\) is the \((d-1)\)-dimensional volume of \(S^{d-1}\). A convex body \(C\) is said to be a translation cover of a family of sets in \(E^d\) if each member of the family lies in some translate of \(C\). Note that, due to Jung's theorem, any set of diameter 1 in \(E^d\) can be covered by a ball of diameter \(\sqrt{{2d \over d+1}}\). The authors prove the following stronger result: The ball of diameter \(\sqrt{{2d \over d+1}}\) is the unique translation cover of minimum mean width for all sets of diameter one in \(E^d\).