Result 328, Functional analysis

Nonexpansive fixed points in reflexive Banach spaces

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 formalization Proof

The bigger picture

Why it matters

A map that never increases distances need not obviously leave any point unchanged. The manuscript claims that a broad class of spaces, including infinite-dimensional ones, nevertheless guarantees such a point without strengthening the geometry of the norm.

What changes?

The manuscript reports that every nonexpansive selfmap, a distance-nonincreasing map from a set to itself, has a fixed point: a point it leaves unchanged. The set must be nonempty, closed, bounded and convex, containing every line segment between its points, in a real reflexive Banach space. Banach spaces are complete normed vector spaces; reflexivity means the canonical representation in the double dual adds no vectors. The claim uses the original norm, without assuming uniform convexity.

What does that help mathematicians do?

If established, this would let researchers prove existence for equations requiring a point to equal its image by checking the stated geometric and distance conditions. Uniform convexity would no longer be a necessary hypothesis for that argument. It would also rule out every fixed-point-free nonexpansive selfmap within this class, forcing any counterexample to violate at least one of the theorem's assumptions.

Are there practical applications?

Its immediate value is foundational: it would make reflexivity sufficient for these fixed-point existence arguments under the stated conditions on the set. The claim does not provide an algorithm, guarantee a unique fixed point or say that repeated application of the map converges. Finding or reliably approximating the point would require additional results.

This section was generated by GPT-6 Astra Medium. This explanation is based on the result summary and manuscript abstracts below. This context is separate from OpenAI's source text.

Manuscript

Fixed Points of Nonexpansive Maps in Reflexive Banach Spaces

September 24, 2026 19 pages Main result formalized in Lean

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.

PDF LaTeX source
Cite (BibTeX) 
@misc{OAI:Fixed-Points-of-Nonexpansive-Maps-in-Reflexive-Banach-Spaces-September-24-2026,
  author = {{OpenAI}},
  title = {{Fixed Points of Nonexpansive Maps in Reflexive Banach Spaces}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Fixed-Points-of-Nonexpansive-Maps-in-Reflexive-Banach-Spaces-September-24-2026/paper.pdf}{OAI:Fixed-Points-of-Nonexpansive-Maps-in-Reflexive-Banach-Spaces-September-24-2026}},
  year = {2026}
}

Lean formalization

OpenAI's note on what the formalization covers, from lean/docs/328.md.

Nonexpansive fixed points in reflexive Banach spaces

The following describes the scope of the Lean formalization related to the following accompanying paper(s):

Scope

The fixed-point question asks whether reflexivity suffices for nonexpansive maps on bounded convex sets. The formalized result gives an affirmative answer: every nonexpansive self-map of a nonempty closed bounded convex subset of a real reflexive Banach space has a fixed point in the original norm. Zero and nonseparable spaces are included, with no uniform convexity, normal structure, or weak-continuity assumption.

Comparator links

Result Comparator statement
Nonexpansive fixed points in reflexive spaces ReflexiveFixedPoints.lean

Data from github.com/openai/math at commit adc7f12, committed October 6, 2026 at 21:58 UTC, last checked for changes about 7 hours ago. Titles, subjects, summaries, abstracts and Lean notes are OpenAI's; page counts are read from the PDFs. The map, related results, search, kinds of results and the named-problem index are Emergent Mind's, built with text embeddings and an LLM, and may contain errors.

An Emergent Mind Labs project. Emergent Mind is not affiliated with OpenAI. None of these results has been peer reviewed. Cite the manuscripts themselves, using the BibTeX on each result's page.