On height orthogonality in normed linear spaces (Q1567137)

Let \(E\) be a real normed linear space of dimension at least two. \textit{C. Alsina}, \textit{P. Guijarro} and \textit{M. S. Tomas} [Rocky Mt. J. Math. 25, No. 3, 843-849 (1995; Zbl 0845.46011)] introduced the following orthogonality: if \(x,y\in E\), then \(x\) is orthogonal to \(y\) in the height sense if either \(\|x\|\cdot\|y\|=0\) or \(\|x-y\|= \|\|y\|\cdot{x\over\|x\|}+\|x\|\cdot{y\over\|y\|}\|\). In the present paper the authors study geometric properties of \(H\)-orthogonality based on a classical property of right triangles.