Approximately bisectrix-orthogonality preserving mappings

TL;DR

This paper introduces a new concept of approximate bisectrix-orthogonality in normed spaces and investigates the properties of linear mappings that approximately preserve this geometric relation.

Contribution

It defines approximate bisectrix-orthogonality and characterizes linear mappings that preserve this relation approximately, extending the understanding of geometric structure preservation in normed spaces.

Findings

Approximate bisectrix-orthogonality is formally defined with bounds depending on epsilon.

Linear approximate similarities preserve the approximate bisectrix-orthogonality relation.

The study provides conditions under which approximate orthogonality is maintained by linear 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 $${x\np{\delta}}_W y\Longrightarrow {Tx \np{\theta}}_W Ty \qquad (x, y\in {\mathcal X})$$ for any $\delta\in[0, 1)$ and certain $\theta\geq 0$.