Nonisomorphic separable Banach spaces at Lipschitz distance one

Abstract

We construct nonisomorphic separable real Banach spaces whose multiplicative Lipschitz distance is one. Building on OpenAI's graph-space construction, we arrange two sequences of graph spaces into countably many blocks and form two spaces that differ by a single additional block. This block distinguishes their linear structures: one space contains a complemented copy of $\ell_1(c_0(\ell_2))$, whereas the other does not. By mapping the additional block to increasingly late positions and rearranging the remaining blocks, we obtain homeomorphisms whose Lipschitz constants and those of their inverses tend to one, while both spaces and their norms remain unchanged.