On the Banach-Mazur distance between continuous function spaces with scattered boundaries

TL;DR

This paper investigates how the Banach-Mazur distance between subspaces of vector-valued continuous functions depends on the scattered structure of their boundaries, providing new bounds and insights into their geometric relationships.

Contribution

It introduces refined bounds for the Banach-Mazur distance based on the scattered structure of boundaries, extending classical results to new classes of function spaces.

Findings

Banach-Mazur distance can be at least 3 when the height of weak peak points exceeds that of the boundary

The estimate improves when the heights are finite and significantly different

New bounds are established for spaces of the form C(K, E)

Abstract

We study the dependence of the Banach-Mazur distance between two subspaces of vector-valued continuous functions on the scattered structure of their boundaries. In the spirit of a result of Gordon, we show that the constant $2$ appearing in the Amir-Cambern theorem may be replaced by $3$ for some class of subspaces. This we achieve by showing that the Banach-Mazur distance of two function spaces is at least 3, if the height of the set of weak peak points of one of the spaces is larger than the height of a closed boundary of the second space. Next we show that this estimate can be improved, if the considered heights are finite and significantly different. As a corollary, we obtain new results even for the case of $\mathcal{C}(K, E)$ spaces.