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

Tomasz Szostok

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.
• Revised: 2014-02-15.
• Accepted: 2014-03-07.
• Published online: 2014-11-12. 