Functional analysis·OpenAI Math·Result 328·Lean formalisation listed
Nonexpansive fixed points in reflexive Banach spaces
One of the 372 results in OpenAI's public mathematics catalogue, filed under functional analysis: 1 preprint, the latest dated 24 September 2026. This page quotes OpenAI's own summary of the claim, links the paper and abstract, and embeds the animation.
In one line: OpenAI reports a result titled ‘Nonexpansive fixed points in reflexive Banach spaces’ (functional analysis) in a preprint dated 24 September 2026. It has not been peer reviewed. The channel's animation, Every OpenAI Math Result, Part 4 of 4: L = BPL, Integer Multiplication and 82 More, is on YouTube.
Chapter 73 of 84 in Every OpenAI Math Result, Part 4 of 4: L = BPL, Integer Multiplication and 82 More · starts at 25:02 · watch from there on YouTube →
What OpenAI claims
Resolves Kirk's reflexive-space fixed-point problem: every nonexpansive selfmap of a nonempty closed bounded convex subset of a real reflexive Banach space has a fixed point. The result uses the original norm, without assuming uniform convexity. (Lean)
Paper
Fixed Points of Nonexpansive Maps in Reflexive Banach Spaces
OpenAI's abstract Every nonexpansive selfmap of a nonempty closed bounded convex subset of a real reflexive Banach space has a fixed point. This resolves the reflexive-space fixed point problem for the given norm.
Status
Preprint. The paper has not been independently reviewed. Treat the result as OpenAI's claim until other mathematicians have checked it. If a video exists, it illustrates the statement; it is not the proof.
Quick answers
- Is it proven?
- Not independently yet. OpenAI posted a preprint on 24 September 2026; it has not been peer reviewed, so treat the result as OpenAI's claim until other mathematicians have checked it.
- What exactly does OpenAI claim?
- Resolves Kirk's reflexive-space fixed-point problem: every nonexpansive selfmap of a nonempty closed bounded convex subset of a real reflexive Banach space has a fixed point. The result uses the original norm, without assuming uniform convexity. (Lean)
- Where is the proof?
- In the paper, Fixed Points of Nonexpansive Maps in Reflexive Banach Spaces (PDF).
- Is there a video?
- Yes, on YouTube (29:10, chapter 73).
Also in
- Chapter 9 of 10 in Tingley's Problem, Separable Quotient Problem and 8 More in Functional Analysis · starts at 2:47 · watch from there on YouTube →
The video
Every OpenAI Math Result, Part 4 of 4: L = BPL, Integer Multiplication and 82 More · Compilation · 29:10 · chapter 73 from 25:02 · description, sources and chapters →
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