The generalized sine function and geometrical properties of normed spaces

Abstract

Let (X, k · k) be a normed space. We deal here with a function s : X × X ! R given by the formula s(x, y) := inf 2R kx + yk kxk (for x = 0 we must define it separately). Then we take two unit vectors x and y such that y is orthogonal to x in the Birkhoff-James sense. Using these vectors we construct new functions x,y which are defined on R. If X is an inner product space, then x,y = sin and, therefore, one may call this function a generalization of the sine function. We show that the properties of this function are connected with geometrical properties of the normed space X.