A study of symmetric points in Banach spaces

TL;DR

This paper provides a complete characterization of symmetric points in Banach spaces, including their behavior in sums of spaces and specific three-dimensional polyhedral examples.

Contribution

It offers a comprehensive description of symmetric points in Banach spaces and their sums, with explicit examples in three-dimensional polyhedral spaces.

Findings

Complete characterization of symmetric points in Banach spaces.

Characterization of symmetric points in sums of Banach spaces.

Explicit identification of symmetric points in specific polyhedral spaces.

Abstract

We completely characterize the left-symmetric points, the right-symmetric points, and, the symmetric points in the sense of Birkhoff-James, in a Banach space. We obtain a complete characterization of the left-symmetric (right-symmetric) points in the infinity sum of two Banach spaces, in terms of the left-symmetric (right-symmetric) points of the constituent spaces. As an application of this characterization, we explicitly identify the left-symmetric (right-symmetric) points of some well-known three-dimensional polyhedral Banach spaces.