Non-expansive maps between unit balls that send the sphere onto the sphere are linear isometries
Kaarel August Kurik
math.FA
Oct 20, 2026 · v1
TL;DR
The main theorem and the key rigidity lemma were formally proved in Lean, with statements and proofs hosted on the Prove2Me platform.
Abstract
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 $L^1[0,1]$.
Problem
The question is whether every non-expansive map between closed unit balls of real Banach spaces that maps the unit sphere into the target sphere, and covers it, must be a linear isometry. This bears on the plasticity problem for unit balls and on a question of Bargetz, Dymond and Pirk about extremality of surjective isometries.
Approach
The map F is shown to be Fréchet differentiable at 0, with a surjective linear isometry T as its derivative. A quantitative rigidity lemma is then proved for non-expansive self-maps H of B_X satisfying ||a+H(a)||=2 on the sphere, using the Baillon-Bruck rate of asymptotic regularity for Krasnosel'skii-Mann iterations. This lemma gives T^{-1}F = id. Theorem 1 (for nontrivial spaces) and Lemma 2 were formalized in Lean.
Results
Every such map is the restriction of a surjective linear isometry. Consequently, plasticity of the unit ball reduces to showing that non-expansive bijections preserve norm 1, and the identity is extremal among non-expansive self-maps for every Banach space. Together with a result of Bargetz, Dymond and Pirk, this shows all surjective isometries of B_X are extremal, including for L^1[0,1].
