Geodesic
On the Banach-Mazur distance between continuous function spaces with scattered boundaries
Czechoslovak Mathematical Journal, Tome 73 (2023) no. 2, pp. 367-393

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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.
DOI : 10.21136/CMJ.2023.0220-21
Classification : 46A55, 46B03, 46E40
Keywords: function space; vector-valued Amir-Cambern theorem; scattered space; Banach-Mazur distance; closed boundary 
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     title = {On the {Banach-Mazur} distance between continuous function spaces with scattered boundaries},
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Rondoš, Jakub. On the Banach-Mazur distance between continuous function spaces with scattered boundaries. Czechoslovak Mathematical Journal, Tome 73 (2023) no. 2, pp. 367-393. doi: 10.21136/CMJ.2023.0220-21

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