TY  - JOUR
ID  - Szostok2015
LB  - Szostok2015
AU  - Szostok, Tomasz
TI  - The generalized sine function and geometrical properties of normed spaces
ST  - The generalized sine function and geometrical properties of normed spaces
JO  - Opuscula Math.
JA  - Opuscula Math.
JF  - Opuscula Mathematica
PY  - 2015
VL  - 35
IS  - 1
SP  - 117
EP  - 126
DO  - http://dx.doi.org/10.7494/OpMath.2015.35.1.117
UR  - http://dx.doi.org/10.7494/OpMath.2015.35.1.117
AB  - 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 \(X\).
KW  - geometry of normed spaces
KW  - smoothness
KW  - strict convexity
KW  - Birkhoff-James orthogonality
KW  - conditional functional equations
SN  - 1232-9274
ER  -