Non-expansive bijections to the unit ball of ${\ell }_1$-sum of strictly convex Banach spaces

Abstract

Extending recent results by Cascales, Kadets, Orihuela and Wingler (2016), Kadets and Zavarzina (2017), and Zavarzina (2017) we demonstrate that for every Banach space $X$ and every collection $Z_i,i\in I$ of strictly convex Banach spaces every non-expansive bijection from the unit ball of $X$ to the unit ball of sum of $Z_i$ by ${\ell }_1$ is an isometry.