This is an announcement for the paper “Non-expansive bijections between unit balls of Banach spaces” by Olesia Zavarzinahttps://arxiv.org/find/math/1/au:+Zavarzina_O/0/1/0/all/0/1.
Abstract: It is known that if $M$ is a finite-dimensional Banach space, or a strictly convex space, or the space $\ell_1$, then every non-expansive bijection $F: B_M\rightarrow B_M$ is an isometry. We extend these results to non-expansive bijections $F: B_E\rightarrow B_M$ between unit balls of two different Banach spaces. Namely, if $E$ is an arbitrary Banach space and $M$ is finite-dimensional or strictly convex, or the space $\ell_1$ then every non-expansive bijection $F: B_E\rightarrow B_M$is an isometry.
The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1704.06961