Approximately bisectrix-orthogonality preserving mappings

Abstract

Regarding the geometry of a real normed space $\mathcal{X}$, we mainly introduce a notion of approximate bisectrix-orthogonality on vectors $x,y\in \mathcal{X}$ as follows: $${x\np{\varepsilon}}_W y \mbox{if and only if} \sqrt{2}\frac{1-\varepsilon}{1+\varepsilon}\|x\|\,\|y\|\leq \Big\|\,\|y\|x+\|x\|y\,\Big\|\leq\sqrt{2}\frac{1+\varepsilon}{1-\varepsilon}\|x\|\,\|y\|.$$ We study class of linear mappings preserving the approximately bisectrix-orthogonality ${\np{\varepsilon}}_W$. In particular, we show that if $T:\mathcal{X}\to \mathcal{Y}$ is an approximate linear similarity, then $${x\npδ}_W y\Longrightarrow {Tx \npθ}_W Ty \qquad (x, y\in {\mathcal X})$$ for any $\delta \in [0,1)$ and certain $\theta \ge 0$.