Czechoslovak Mathematical Journal, Vol. 73, No. 2, pp. 367-393, 2023
On the Banach-Mazur distance between continuous function spaces with scattered boundaries
Jakub Rondoš
Received June 18, 2021. Published online March 16, 2023.
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 Y. Gordon (1970), we show that the constant 2 appearing in the Amir-Cambern theorem may be replaced by 3 for some class of subspaces. We achieve this 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 differs from 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.
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Affiliations: Jakub Rondoš, Department of Mathematical Analysis, Faculty of Mathematics and Physics, Charles University, Sokolovská 49/83, 186 75 Prague 8, Czech Republic, e-mail: jakub.rondos@gmail.com
