On the numerical index of polyhedral Banach spaces
Debmalya Sain, Kallol Paul, Pintu Bhunia, Santanu Bag

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 -dimensional polyhedral Banach spaces for the first time, in order to illustrate the applicability of our method.
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