Borsuk's covering for blunt bodies
Denote by \(d_ n\) the smallest number such that there exists a covering of the unit sphere \(S^{n-1}\) by \(n+1\) sets each of diameter \(\leq d_ n\). (Thus \(d_ 2=2\pi /3).)\) Let C be a convex body in \(E^ n\) and let \(x\in \partial C\). Denote by \(S(x,C)\subset S^{n-1}\) the set of directions of all diameters of C starting from x. Suppose that the circumradius of \(S(x,C)\) in \(S^{n-1}\) is \(<(\pi -d_ n)\) for any \(x\in \partial C\) where \(S(x,C)\neq \emptyset\). Then C can be covered by \(n+1\) sets, each of diameter less than diameter \(C.\)
scientific article; zbMATH DE number 4022043
scientific article