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

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 spaces.