Convex and metric geometry·OpenAI Math·Result 098·Lean formalisation listed
Compact counterexamples to bi-Lipschitz dimension reduction
One of the 372 results in OpenAI's public mathematics catalogue, filed under convex and metric geometry: 1 preprint, the latest dated 25 September 2026. This page quotes OpenAI's own summary of the claim, links the paper and abstract, and embeds the animation.
In one line: OpenAI reports a result titled ‘Compact counterexamples to bi-Lipschitz dimension reduction’ (convex and metric geometry) in a preprint dated 25 September 2026. It has not been peer reviewed. The channel's animation, Every OpenAI Math Result, Part 1 of 4: Goldfeld's Conjecture and 82 More, is on YouTube.
Chapter 64 of 83 in Every OpenAI Math Result, Part 1 of 4: Goldfeld's Conjecture and 82 More · starts at 21:52 · watch from there on YouTube →
What OpenAI claims
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)
Paper
A doubling Hilbert subset with no finite-dimensional bi-Lipschitz embedding
OpenAI's abstract 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.
Status
Preprint. The paper has not been independently reviewed. Treat the result as OpenAI's claim until other mathematicians have checked it. If a video exists, it illustrates the statement; it is not the proof.
Quick answers
- Is it proven?
- Not independently yet. OpenAI posted a preprint on 25 September 2026; it has not been peer reviewed, so treat the result as OpenAI's claim until other mathematicians have checked it.
- What exactly does OpenAI claim?
- 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)
- Where is the proof?
- In the paper, A doubling Hilbert subset with no finite-dimensional bi-Lipschitz embedding (PDF).
- Is there a video?
- Yes, on YouTube (28:48, chapter 64).
Also in
- Chapter 9 of 10 in Petty's Projection Conjecture, Log Brunn–Minkowski Conjecture and 8 More · starts at 2:47 · watch from there on YouTube →
The video
Every OpenAI Math Result, Part 1 of 4: Goldfeld's Conjecture and 82 More · Compilation · 28:48 · chapter 64 from 21:52 · description, sources and chapters →
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