The core of the unit sphere of a Banach space

Abstract

A geometric invariant or preserver is essentially a geometric property of the unit sphere of a real Banach space that remains invariant under the action of a surjective isometry onto the unit sphere of another real Banach space. A new geometric invariant of the unit ball of a real Banach space was introduced and analyzed in this manuscript: the core of the unit sphere. This geometric invariant consists of all points in the unit sphere of a real Banach space, which are contained in a unique maximal face. It is, in a geometrical sense, the opposite of fractal-like sets such as starlike sets. Classical geometric properties, such as smoothness and strict convexity, were employed to characterize the core of the unit sphere. Also, the core was related to a recently introduced new index: the index of strong rotundity. A characterization of the core in terms of the index of strong rotundity was provided. Finally, applications to longstanding open problems, such as Tingley’s problem, were provided by presenting a new notion: Mazur-Ulam classes of Banach spaces.