When locally distance-preserving Banach bijections are global isometries
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Suppose a bijection between real Banach spaces preserves distances between every pair of points in each closed ball of radius 1/4. Must it be a global isometry? The unrestricted question does not assume separability.
Why it matters and what counts as progress
Why it matters: This asks whether a fixed-scale local distance condition forces global metric rigidity.
What counts as progress: A general local-to-global theorem or a bijective counterexample; a positive theorem for separable source spaces does not settle the unrestricted case.
How the problem developed
- 7 Aug 2023 · Related result [1]
Michiya Mori Proved that a locally isometric surjection between Banach spaces is a global isometry when the source space is separable.
- 30 Aug 2026 · Counterexample [2]
Yoshito Ishiki Constructed a bijection between real Banach spaces that preserves distances on every closed ball of radius 1/4 but is not a global isometry.
AI contribution
A counterexample to Scottish Book Problem 155
Yoshito Ishiki constructs real Banach spaces X and Y and a bijection U from X to Y that preserves distances between every pair of points in each closed ball of radius 1/4, but is not a global isometry. This answers the unrestricted Scottish Book Problem 155 negatively.
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