On the numerical index of polyhedral Banach spaces

TL;DR

This paper introduces a new method to estimate the numerical index of finite-dimensional real polyhedral Banach spaces, successfully computing exact values for certain 3D cases and advancing understanding of these geometric properties.

Contribution

The paper presents a general approach to estimate the numerical index using finitely many functionals, and computes exact indices for specific 3D polyhedral Banach spaces.

Findings

Developed a method to estimate the numerical index using finitely many functionals.

Successfully computed the exact numerical index for a family of 3D polyhedral Banach spaces.

Demonstrated the applicability of the method to new classes of spaces.

Abstract

The computation of the numerical index of a Banach space is an intriguing problem, even in case of two-dimensional real polyhedral Banach spaces. In this article we present a general method to estimate the numerical index of any finite-dimensional real polyhedral Banach space, by considering the action of only finitely many functionals, on the unit sphere of the space. We further obtain the exact numerical index of a family of $3$-dimensional polyhedral Banach spaces for the first time, in order to illustrate the applicability of our method.