Local approximate symmetry of Birkhoff-James orthogonality in normed linear spaces

TL;DR

This paper investigates approximate symmetry properties of Birkhoff-James orthogonality in finite-dimensional normed spaces, establishing new results and characterizations, especially in polyhedral Banach spaces, extending previous work on the topic.

Contribution

It introduces new characterizations of approximate symmetry of Birkhoff-James orthogonality, particularly linking it to geometric properties in polyhedral Banach spaces.

Findings

Approximate symmetry holds in all finite-dimensional Banach spaces (Dragomir's sense).

In polyhedral Banach spaces, symmetry is equivalent to a new geometric property.

Extends recent results on global approximate symmetry of Birkhoff-James orthogonality.

Abstract

Two different notions of approximate Birkhoff-James orthogonality in nor\-med linear spaces have been introduced by Dragomir and Chmie\-li\'n\-ski. In the present paper we consider a global and a local approximate symmetry of the Birkhoff-James orthogonality related to each of the two definitions. We prove that the considered orthogonality is approximately symmetric in the sense of Dragomir in all finite-dimensional Banach spaces. For the other case, we prove that for finite-dimensional polyhedral Banach spaces, the approximate symmetry of the orthogonality is equivalent to some newly introduced geometric property. Our investigations complement and extend the scope of some recent results on a global approximate symmetry of the Birkhoff-James orthogonality.