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Non-expansive maps between unit balls that send the sphere onto the sphere are linear isometries

Kaarel August Kurik

Let X and Y be real Banach spaces and F:B_X→B_Y be non-expansive on their closed unit balls. If F(S_X)⊆S_Y⊆F(B_X), then F is the restriction of a surjective real-linear isometry X→Y. Sphere coverage is required; coverage of the ball’s interior and injectivity are not assumed.

Sources for this record

What was already known

Earlier ball-plasticity results covered particular Banach spaces. This theorem reduces plasticity to preservation of the unit sphere and derives extremality of the identity among non-expansive self-maps of every real Banach unit ball. The proof uses a quantitative common-face rigidity lemma and the Baillon–Bruck asymptotic-regularity estimate.

AI involvement AI-led

Kurik explicitly attributes the mathematical results entirely to OpenAI and Anthropic language models, primarily GPT-6 Astra and Claude Opus 5.5. The human author takes responsibility for correctness and significance and reports checking the abstract, bibliography and typesetting. The disclosure establishes model-led mathematical work but does not establish task-only autonomy or a complete human check of the argument.

Importance Notable

Provides a general rigidity criterion and a reduction for the ball-plasticity problem. Its hypotheses differ from Tingley’s sphere-isometry extension problem, which already assumes an isometry of spheres.

Limitations

The theorem does not prove that every non-expansive bijection of a unit ball preserves its sphere, so general ball plasticity remains unresolved. The platform’s displayed formal statements assume nonzero spaces; the paper treats real Banach spaces generally. Complete checking receipts and a named human explanation of the model-generated argument were not established by the reviewed sources.

Technical verification Reported

Reported is the default when evidence for a higher level is insufficient. The paper reports Lean proofs of Theorem 1 and Lemma 2. Prove2Me displays the matching nonzero-Banach-space statement Plasticity.U16 and a quantitative common-face rigidity statement under Lean 4.33.1. Its public pages show proof-omitted statement templates and endorsement labels, but the reviewed pages provide no complete proof execution receipt or dependency/axiom audit. Reported records that checking-evidence gap; it is not a failed-check finding.

Human understanding Few

Few is the default when evidence for a higher level is insufficient. The author attributes the proof to models and vouches for the result. The reviewed sources do not document how completely a named human can explain the central derivative and rigidity mechanism; Few is the documentation default.

Correction and publication history