@Article{Szostok2015,
author = "Szostok, Tomasz",
title = "The generalized sine function and geometrical properties of normed spaces",
journal = "Opuscula Math.",
year = "2015",
volume = "35",
number = "1",
pages = "117--126",
doi = "http://dx.doi.org/10.7494/OpMath.2015.35.1.117",
url = "http://dx.doi.org/10.7494/OpMath.2015.35.1.117",
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 \(X\).",
keywords = "geometry of normed spaces, smoothness, strict convexity, Birkhoff-James orthogonality, conditional functional equations",
issn = "1232-9274"
}