Uniform toroidal separation in complex normed spaces
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Does every infinite-dimensional complex normed space contain unit vectors uniformly separated by more than one even after multiplying either vector by any scalar of modulus one? Completeness is not assumed.
Why it matters and what counts as progress
Why it matters: Toroidal separation strengthens ordinary Elton–Odell separation by identifying all phase rotations of a vector.
What counts as progress: A sequence and ε>0 with the phase-uniform bound 1+ε, or a normed space excluding it, without extra geometric assumptions.
How the problem developed
- 2018 · Open problem [1]
Petr Hájek, Tomasz Kania and Tommaso Russo Formulated the toroidal Elton–Odell question: whether every infinite-dimensional complex normed space contains a sequence uniformly separated modulo multiplication by scalars of modulus one.
- 19 Jul 2026 · Reported claim [2]
Antonio Acuaviva and Pablo Acuaviva Reported a positive answer for every infinite-dimensional complex normed space; the authors say AI systems supplied key ideas and proofs that they verified and refined.
AI contribution
Mathematical Discovery in the Wild: AI-Guided Proofs in Banach Space Theory
Every infinite-dimensional complex normed space has unit vectors separated by at least 1+ε for some ε>0 even after either vector is multiplied by any scalar of modulus one.
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