Linearity of sphere-covering non-expansive maps between Banach balls
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Let F map the closed unit ball of a real Banach space X non-expansively into that of Y. If F sends sphere points to sphere points and its image covers the entire unit sphere of Y, must it extend to a surjective real-linear isometry?
Why it matters and what counts as progress
Why it matters: The condition offers a general rigidity criterion for ball maps beyond special-space plasticity results.
What counts as progress: A rigidity proof or counterexample under exactly the sphere-preservation and sphere-coverage assumptions; injectivity and interior coverage are not assumed.
How the problem developed
- 20 Oct 2026 · Theorem claim [1]
Kaarel August Kurik Claimed that every non-expansive map between real Banach unit balls whose image covers the unit sphere and sends sphere points to sphere points is the restriction of a surjective linear isometry.
AI contribution
Non-expansive maps between unit balls that send the sphere onto the sphere are linear isometries
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.
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Non-expansive maps between unit balls that send the sphere onto the sphere are linear isometries