∑math20

Nonexpansive fixed points in reflexive Banach spaces

1 paperLean

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.

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▶Introduction: the problem and why it mattersin preparation

▶The presented solutionin preparation

Papers 1

Fixed Points of Nonexpansive Maps in Reflexive Banach Spaces

OpenAI · September 24, 2026

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.

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Cite

@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}
}

What was formalized in Lean

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

Fixed Points of Nonexpansive Maps in Reflexive Banach Spaces

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

Scope note on GitHub →