We prove that for any $\ell_\infty$-sum $Z = \bigoplus_{ i \in [n]} X_i$ of finitely many strictly convex Banach spaces $(X_i)_{ i \in [n]}$, an extremeness preserving 1-Lipschitz bijection $f\colon B_Z \to B_Z$ is an isometry, by constraining the componentwise behavior of the inverse $g=f^{ -1}$ with a theorem admitting a graph-theoretic interpretation. We also show that if $X, Y$ are Banach spaces, then a bijective 1-Lipschitz non-isometry of type $B_X \to B_Y$ can be used to construct a bijective 1-Lipschitz non-isometry of type $B_{ X'} \to B_{ X'}$ for some Banach space $X'$, and that a homeomorphic 1-Lipschitz non-isometry of type $B_X \to B_X$ restricts to a homeomorphic 1-Lipschitz non-isometry of type $B_S \to B_S$ for some separable subspace $S \leq X$.
Categories: math.FA