A study on preservation of parallel pairs and triangle equality attainment pairs

TL;DR

This paper explores the structure and preservation of parallel pairs and triangle equality attainment pairs in normed linear spaces, providing characterizations and conditions under which these pairs are preserved by operators, with implications for isometries.

Contribution

It offers new functional characterizations of these pairs, analyzes their preservation under operators, and characterizes isometries in certain polyhedral Banach spaces.

Findings

Preservation of TEA pairs is equivalent to preservation of parallel pairs in finite-dimensional Banach spaces.

Characterization of isometries as norm-one bijective operators preserving extreme points.

Structural insights into the geometry of normed spaces related to these pairs.

Abstract

This paper investigates the structure and preservation of parallel pairs and triangle equality attainment (TEA) pairs in normed linear spaces. We begin by providing functional characterizations of these pairs in normed linear spaces and provide a characterization of finite-dimensional Banach spaces with numerical index one. We then study parallel pairs and TEA pairs in the $p$-direct sum of normed linear spaces. Next, we show that in finite-dimensional Banach spaces, a bijective bounded linear operator preserves TEA pairs if and only if it preserves parallel pairs, and this equivalence remains valid in finite-dimensional polyhedral spaces for operators with rank greater than one. We further explore some geometric consequences related to this preservation and provide a characterization of isometries on certain polyhedral Banach spaces as norm-one bijective operators that preserve the norm at extreme points and also preserve the parallel or TEA pairs.