Result 098, Convex and metric geometry

Compact counterexamples to bi-Lipschitz dimension reduction

Every infinite-dimensional real Banach space contains a compact doubling set that admits no bi-Lipschitz embedding into any finite-dimensional normed space. The doubling constant is universal. This answers the Lang–Plaut problem negatively, even for compact subsets of Hilbert space.

Lean formalization Disproof or counterexample

The bigger picture

Why it matters

A set can look low-dimensional at every scale yet resist every finite-dimensional representation that preserves distances within a fixed multiplicative factor. The manuscript reports that this obstruction occurs even for compact subsets of Hilbert space.

What changes?

The claim covers every infinite-dimensional real Banach space, a complete normed vector space. Each contains a compact doubling set: every sequence has a subsequence converging within the set, and every ball can be covered by a bounded number of balls of half its radius. That number has a universal bound, independent of the ambient space. Nevertheless, no map into any finite-dimensional real normed space preserves all pairwise distances up to fixed positive multiplicative bounds, the requirement for a bi-Lipschitz embedding.

What does that help mathematicians do?

Researchers therefore cannot deduce finite-dimensional embeddability from a uniform doubling bound, even after adding compactness and assuming the set lies in Hilbert space. The obstruction is not merely a demand for more dimensions or larger distortion: the reported sets defeat every finite dimension and every finite distance-distortion bound. This gives a negative answer to the Lang-Plaut problem.

Are there practical applications?

The immediate value is foundational: it identifies a boundary for dimension reduction that aims to retain all distances. A general finite-dimensional embedding theorem for compact doubling sets would need additional restrictions or weaker distance-preservation requirements. The result concerns entire compact sets, so it does not by itself rule out useful representations of finite samples.

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

A doubling Hilbert subset with no finite-dimensional bi-Lipschitz embedding

September 25, 2026 13 pages Main result formalized in Lean

Every infinite-dimensional real Banach space contains a compact doubling subset that admits no bi-Lipschitz embedding into any finite-dimensional real normed space. The doubling constant has a universal bound, independent of the ambient Banach space. This answers the Lang–Plaut problem negatively, even for compact subsets of Hilbert space.

PDF LaTeX source

Cite (BibTeX)

@misc{OAI:A-doubling-Hilbert-subset-with-no-finite-dimensional-bi-Lipschitz-embedding-September-25-2026,
  author = {{OpenAI}},
  title = {{A doubling Hilbert subset with no finite-dimensional bi-Lipschitz embedding}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/A-doubling-Hilbert-subset-with-no-finite-dimensional-bi-Lipschitz-embedding-September-25-2026/main.pdf}{OAI:A-doubling-Hilbert-subset-with-no-finite-dimensional-bi-Lipschitz-embedding-September-25-2026}},
  year = {2026}
}

Lean formalization

OpenAI's note on what the formalization covers, from lean/docs/098.md.

Compact counterexamples to bi-Lipschitz dimension reduction

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

A doubling Hilbert subset with no finite-dimensional bi-Lipschitz embedding

Scope

The formalized result gives a doubling subset of real $\ell_2$ , with doubling constant at most $76800$ , that admits no bi-Lipschitz embedding into any finite-dimensional Euclidean space at any finite distortion. A companion gives one universal doubling constant such that every infinite-dimensional real Banach space contains a compact doubling subset with no bi-Lipschitz embedding into any finite-dimensional normed space.

Comparator links

Result	Comparator statement
Doubling Hilbert subset without finite-dimensional embedding	DoublingHilbert.lean
Compact counterexamples in infinite-dimensional Banach spaces	CompactBanach.lean

At a glance

Subject: Convex and metric geometry
Manuscripts: 1
Pages: 13
Dated: Sep 25, 2026
Lean formalization: Yes
Reasoning summary: No

Named problems addressed

Lang–Plaut problemDisproved

Indexed by Emergent Mind with an LLM from the text above; check the manuscripts for the precise statements.

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