Non-expansive maps between unit balls that send the sphere onto the sphere are linear isometries
Let $X$, $Y$ be real Banach spaces with closed unit balls $B_X$, $B_Y$ and unit spheres $S_X$, $S_Y$. We prove that every non-expansive map $F\colon B_X\to B_Y$ with $F(S_X)\subseteq S_Y\subseteq F(B_X)$ is the restriction of a surjective linear isometry $X\to Y$. Consequently, a non-expansive bijection of $B_X$ onto itself that fails to be an isometry must map some point of $S_X$ into the open unit ball, so the plasticity of $B_X$ reduces to showing that non-expansive bijections preserve norm 1. The proof shows that $F$ is Fréchet differentiable at $0$ with a surjective linear isometry $T$ as its derivative, and then proves $T^{-1}F=\mathrm{id}$ using a quantitative rigidity lemma for non-expansive self-maps $H$ of $B_X$ satisfying $\|a+H(a)\|=2$ for $a\in S_X$; the latter relies on the Baillon-Bruck rate of asymptotic regularity for Krasnosel'skii-Mann iterations. As a corollary, the identity is extremal in the set of non-expansive self-maps of $B_X$ for every Banach space $X$, which together with a result of Bargetz, Dymond and Pirk shows that all surjective isometries of $B_X$ are extremal, answering a question of theirs, including the case of $L^1[0,1]$.