The generalized sine function and geometrical properties of normed spaces

Abstract

Let $(X,|\cdot|)$ be a normed space. We deal here with a function $s:X\times X\to\mathbb{R}$ given by the formula $s(x,y):=\inf_{\lambda\in\mathbb{R}}\frac{|x+\lambda y|}{|x|}$ (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 $\phi_{x,y}$ which are defined on $\mathbb{R}$. If $X$ is an inner product space, then $\phi_{x,y}=\sin$ 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 $X$.