Opuscula Math. 35, no. 1 (2015), 117-126
http://dx.doi.org/10.7494/OpMath.2015.35.1.117
Opuscula Mathematica
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 \(X\).
Keywords: geometry of normed spaces, smoothness, strict convexity, Birkhoff-James orthogonality, conditional functional equations.
Mathematics Subject Classification: 46B20, 39B55, 39B52.
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- Tomasz Szostok
- Silesian University, Institute of Mathematics, Bankowa 14, 40-007 Katowice, Poland
- Communicated by Karol Baron.
- Received: 2013-11-29.
- Revised: 2014-02-15.
- Accepted: 2014-03-07.
- Published online: 2014-11-12.
