Orthogonality induced by norm derivatives : A new geometric constant and symmetry

TL;DR

This paper introduces a new geometric constant to compare different types of orthogonality in normed spaces, characterizes inner product spaces, and describes symmetric elements with respect to $ ho$-orthogonality.

Contribution

It defines the constant $ extGamma( ext{X})$ to analyze $ ho$-orthogonality, characterizes inner product spaces via symmetry properties, and describes symmetric elements in specific Banach spaces.

Findings

The constant $ extGamma( ext{X})$ relates to geometric properties of the space.

Inner product spaces are characterized by symmetric $ ho$-orthogonality.

Complete descriptions of symmetric elements in certain Banach spaces.

Abstract

In this article we study the difference between orthogonality induced by the norm derivatives (known as $\rho$-orthogonality) and Birkhoff-James orthogonality in a normed linear space $ \mathbb X$ by introducing a new geometric constant, denoted by $\Gamma(\mathbb{X}).$ We explore the relation between various geometric properties of the space and the constant $\Gamma(\mathbb{X}).$ We also investigate the left symmetric and right symmetric elements of a normed linear space with respect to $\rho$-orthogonality and obtain a characterization of the same. We characterize inner product spaces among normed linear spaces using the symmetricity of $\rho$-orthogonality. Finally, we provide a complete description of both left symmetric and right symmetric elements with respect to $\rho$-orthogonality for some particular Banach spaces.