Conditional plasticity of the unit ball of the _∞-sum of finitely many strictly convex Banach spaces

We prove that for any _∞-sum Z = i ∈ [n] Xᵢ of finitely many strictly convex Banach spaces (Xᵢ)i ∈ [n], an extremeness preserving 1-Lipschitz bijection f BZ → BZ 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 BX → BY can be used to construct a bijective 1-Lipschitz non-isometry of type BX' → BX' for some Banach space $X'$, and that a homeomorphic 1-Lipschitz non-isometry of type BX → BX restricts to a homeomorphic 1-Lipschitz non-isometry of type BS → BS for some separable subspace S ≤ X.