
Journal of Convex Analysis 21 (2014), No. 4, 951964 Copyright Heldermann Verlag 2014 Weak Concavity of the Antidistance Function Maxim V. Balashov Department of Higher Mathematics, Moscow Institute of Physics and Technology, Institutskii pereulok 9, Dolgoprudny, Moscow region, Russia 141700 balashov73@mail.ru Maxim O. Golubev Department of Higher Mathematics, Moscow Institute of Physics and Technology, Institutskii pereulok 9, Dolgoprudny, Moscow region, Russia 141700 maksimkane@mail.ru We prove that the function which for a given point of a Banach space gives the largest distance to the points of a given convex closed bounded set (antidistance) is weakly concave on the complement to some neighborhood of the set if and only if the set is a summand of some ball of some radius. We obtain precise estimates for parameters of weak concavity via the size of the neighborhood and radius of the ball in the Hilbert space. Keywords: Antidistance function, uniform convexity, uniform smoothness, weak concavity, modulus of weak concavity, proximal smoothness, Psupporting condition. MSC: 49J52, 58C20, 52A07; 26B25, 52A41, 52A05 [ Fulltextpdf (154 KB)] for subscribers only. 