On the minimal convex annulus of a planar convex body
Let \(K\) be a fixed smooth and strictly convex body in \(E^ 2\) with 0 in its interior. By a convex annulus with centre \(c\) we mean the set of all points inside the convex body of the form \(\rho K+c\) and outside of the form \(\sigma K+c\), where \(0\leq\sigma\leq\rho\). A minimal convex annulus \({\mathcal K}(C)\) of a convex body \(C\) is a convex annulus containing the boundary of \(C\) and for which \(|\rho-\sigma|\) is minimal. We first generalize results of Bonnesen, Kritikos and Barany on ``circular'' annulus by proving the existence of a Radon partition of the set of contact points of the boundary of \({\mathcal K}(C)\) and \(C\). Subsequently the uniqueness of \({\mathcal K}(C)\) is shown. Finally we extend a result of the second author on circular annulus showing that for a typical convex body \(C\) (typical in the sense of Baire categories) the minimal convex annulus meets the boundary of \(C\) in precisely four points.
scientific article; zbMATH DE number 164918