The Lang-Plaut problem: does every doubling subset of Hilbert space bi-Lipschitz embed in some Euclidean space?

A metric space is doubling if every ball is covered by a bounded number of balls of half the radius. Every subset of $\mathbb R^k$ is doubling, and Assouad's theorem embeds every doubling space bi-Lipschitzly into some $\mathbb R^k$ after snowflaking the metric to $d^\alpha$ , $\alpha<1$ . For $L_p$ with $p>2$ , Lafforgue-Naor and Bartal-Gottlieb-Neiman found doubling subsets with no such embedding. Lang and Plaut asked the Hilbert case for the original metric, and Gupta, Krauthgamer and Lee asked it independently for algorithmic dimension reduction. Does every doubling subset of $\ell_2$ admit a bi-Lipschitz embedding into $\mathbb R^k$ for some finite $k$ and finite distortion?

Result: Disproved(see note)
Status: Candidate (review pending)
AI contribution: AI-discovered
Method: Construction
Field: Metric geometry; bi-Lipschitz embeddings and dimension reduction
Posed by: Urs Lang and Conrad Plaut, Bilipschitz embeddings of metric spaces into space forms, Geom. Dedicata 87 (2001), Question 2.4; independently Gupta, Krauthgamer and Lee (FOCS 2003)
Year posed: 2001
Years open: 25y
Solved: 2026-09-25
Model: Unreleased internal OpenAI model
Vendor: OpenAI
Collaborators: —
Verification: Lean-checked, statement unaudited
Publication: Announced
Collection: OpenAI math release (October 2026), version adc7f12
Significance: 38 / 100
Disclosed cost: —
Wikipedia: No dedicated article

What was actually shown

Claims a negative answer: a fixed doubling subset of $\ell_2$ (doubling constant at most 76800) with no bi-Lipschitz embedding into any $\mathbb R^k$ ; more generally every infinite-dimensional real Banach space contains a compact set of universally bounded doubling constant with no bi-Lipschitz embedding into any finite-dimensional normed space, and the same metric embeds isometrically in every $L_p$ , $1\le p<\infty$ . It does NOT bear on snowflaked embeddings (Assouad). Priority note: Schioppa announced a negative answer in 2017 (arXiv:1703.10265) but withdrew it.

What the AI did

The release README says all results were produced by an unreleased internal OpenAI model, the vast majority by one fixed procedure using on average about three hours of ChatGPT Pro thinking compute per result. Its named exceptions (the Hodge conjecture for CM abelian varieties, and the Re(s) > 11/12 zero-free region, whose write-up was human edited for readability) do not concern this family. The manuscripts are credited to OpenAI with no human author named. The single manuscript constructs the counterexample and its Banach-space extension; both main statements are formalized in Lean in the release catalogue.

Verification

No independent mathematician has checked this yet. Theorem 1.1 and the abstract's Banach-space statement were read against Question 2.4 of Lang-Plaut: a fixed subset of real $\ell_2$ with doubling constant at most 76800 admits no bi-Lipschitz embedding into any $\mathbb R^k$ at any distortion. Both are in formalization.yaml. ComparatorChallenges/DoublingHilbert.lean (OAI.DoublingHilbert.main, OAI/Geometry/DoublingHilbert/Main.lean) states exactly Theorem 1.1, with doubling defined by open balls centered in the subset and embeddings allowing any scale and distortion. ComparatorChallenges/CompactBanach.lean (OAI.CompactBanach.main) states the stronger form: one constant $\Lambda$ such that every infinite-dimensional real Banach space contains a compact $\Lambda$ -doubling set with no bi-Lipschitz embedding into any finite-dimensional normed space. Permitted axioms propext, Quot.sound, Classical.choice. Not rebuilt here.

The Lang-Plaut problem: does every doubling subset of Hilbert space bi-Lipschitz embed in some Euclidean space?

What was actually shown

What the AI did

Verification

Sources

Discussion