OpenAI · 098-01 · 2026-09-25 · Convex and metric geometry
A doubling Hilbert subset with no finite-dimensional bi-Lipschitz embedding
preprints/A-doubling-Hilbert-subset-with-no-finite-dimensional-bi-Lipschitz-embedding-September-25-2026
Family 098. A negative answer to the Lang–Plaut problem.
The authors are OpenAI. This companion does not verify the proof and does not adopt it as a theorem of Ngo Cheung’s series. It is not a premise. O.1 stays NOT_CLOSED. Anti-de Sitter results are not a de Sitter state, not Em02-E’s relational lapse, not XVI-P’s QdS scheme, and not XVI-Q’s sigma-image record. No quantum coefficient is admitted. No compilation is asserted.
Read against the standing problems
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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.
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.
Preprint source and Lean scope
The abstract above is the catalogue abstract. The sections below are the full preprint source, 0 sections. This page is not only the abstract. The manuscript file is loading.
This preprint is named in the Lean scope note for 098: Compact counterexamples to bi-Lipschitz dimension reduction. The authors are OpenAI. This companion does not verify the formalization. It is not a premise. The OAI library is not imported. No compilation is asserted. O.1 stays NOT_CLOSED.