On real nonisomorphic Banach spaces with homeomorphic geometric structure spaces (Q6564351)

Let \(X\) be a real Banach space and let \(B_X\) denote the closed unit ball. On the collection of all maximal faces \(F\) of \(B_X\), consider the set-valued function \(\Phi^\ast(F) = \{f \in B_{X^\ast}: f=1\text{ on }F\}\). The author has introduced in an earlier work [J. Math. Anal. Appl. 505, No.~1, Article ID 125444, 12~p. (2022; Zbl 1486.46017)], the notion of a geometric space associated with \(X\), \({\mathcal O}(X)= \bigcup \{\ker(f): f\in \Phi^\ast(F)\}\). A result from [loc. cit.] shows that Banach spaces which are equivalent with respect to, Birkhoff-James orthogonality have homeomorphic geometric structure space. This interesting paper gives a counterexample for the converse implication.\N\NStarting with an example due to \N\textit{P.~Koszmider} [Math. Ann. 330, No.~1, 151--183 (2004; Zbl 1064.46009)]\Nof a compact set \(K\) for which the space of continuous functions \(C(K)\) is not isomorphic to any of its hyperplanes, for a non-atomic probability measure \(\mu\) on \(K\) and for the hyperplane \(\ker(\mu)\), the author shows that the geometric structure spaces are homeomorphic but \(\ker(\mu)\) contains no non-zero right symmetric points of Birkhoff-James orthogonality and hence is not equivalent to \(C(K)\) in this sense.