The fixed point property for nonexpansive maps in reflexive Banach spaces
A Banach space has the fixed point property for nonexpansive maps if every with , on every nonempty closed bounded convex set , has a fixed point, nonexpansiveness being measured in the given norm. Browder and Gohde (1965) proved it for uniformly convex spaces and Kirk (1965) for reflexive spaces when the domain has normal structure; Karlovitz showed normal structure is not necessary, Maurey proved it for reflexive subspaces of , and Alspach (1981) gave a fixed-point-free nonexpansive map on a weakly compact convex subset of , so weak compactness alone does not suffice. Dominguez Benavides showed that every reflexive space has an equivalent norm with the property. Does every real reflexive Banach space have the fixed point property for nonexpansive maps in its original norm?
- Result
- Proved(see note)
- Status
- Candidate (review pending)
- AI contribution
- AI-discovered
- Method
- Argument
- Field
- Functional analysis: metric fixed point theory
- Posed by
- Classical question of metric fixed point theory arising from Kirk's 1965 theorem; the manuscript calls it the reflexive-space fixed point problem and names no poser
- Year posed
- —
- Years open
- —
- Solved
- 2026-09-24
- Model
- Unreleased internal OpenAI model
- Vendor
- OpenAI
- Collaborators
- —
- Verification
- Lean-checked, statement unaudited
- Publication
- Announced
- Collection
- OpenAI math release (October 2026), version adc7f12
- Significance
- 45 / 100
- Disclosed cost
- —
- Wikipedia
- No dedicated article
What was actually shown
Theorem 1.1: if is a real reflexive Banach space and is nonempty, closed, bounded and convex, every nonexpansive has a fixed point, with no uniform convexity, normal structure or separability assumed and the norm unchanged. Corollary: for any commuting family of nonexpansive self-maps of such , the common fixed set is nonempty and a nonexpansive retract (via Bruck 1974). The proof argues by contradiction from a minimal invariant set, using the Goebel-Karlovitz lemma, an adaptive anchor, a tree of approximate fixed point images and a weighted switching estimate. It says nothing about nonreflexive spaces with the weak fixed point property, and gives no constructive iteration.
What the AI did
The OpenAI math release (github.com/openai/math, commit adc7f12) states that its results were produced by an unreleased internal OpenAI model under one fixed procedure, averaging about three hours of ChatGPT Pro thinking compute per result, across roughly 4,000 posed problems; outputs were then grouped into families and filtered for significance. This result is not among the README's stated exceptions (the Riemann zeta zero-free region work and the Hodge conjecture for CM abelian varieties). The manuscript is credited to OpenAI alone and names no human author.
Verification
No independent mathematician has checked this yet. Checked here: the introduction and Theorem 1.1 of the TeX source; the proof was not refereed. Lean-checked on the Comparator challenge ReflexiveFixedPoints, listed in the release's formalization catalogue (OAI.ReflexiveFixedPoints.exists_fixedPoint_of_canonicallyReflexive, OAI/Analysis/Nonexpansive/Main.lean). Its statement: in a complete real normed space whose canonical map into the bidual is surjective, every map of a nonempty closed bounded convex set into itself that is nonexpansive in the given norm has a fixed point. That is the headline claim, with no separability or other geometric hypothesis. Permitted axioms: propext, Quot.sound, Classical.choice. The statement was read here; the development was not rebuilt here.