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    Item type:Article, Access status: Open Access ,
    The generalized sine function and geometrical properties of normed spaces
    (2015) Szostok, Tomasz
    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 $X$.