Result 094, Convex and metric geometry

Subpolynomial dimension reduction in L_p

For every fixed $1\lt p\lt \infty$ and distortion D > 1, every n-point subset of real L_p embeds into $\ell_p^d$ with distortion at most D and dimension $d=n^{o(1)}$ , answering Naor's sublinear-dimension question for p ≠ 2. In contrast, exact embeddings require worst-case dimension $\Theta(n^2)$ when p ≠ 2.

Lean formalization Proof

The bigger picture

Why it matters

Allowing a small error in distances can dramatically reduce the coordinates needed to represent a finite geometric configuration. This manuscript claims such a reduction for every finite collection of points in real Lp spaces.

What changes?

For every fixed p strictly between 1 and infinity and fixed D greater than 1, the manuscript reports an embedding of any n points in real Lp into d real coordinates with the same exponent p. Lp measures size by integrating pth powers and taking a pth root. After rescaling, distances change by factors between 1 and D. For these fixed parameters, d grows more slowly than every positive power of n. The map need not be linear.

What does that help mathematicians do?

For p other than 2, exact distance preservation requires dimension proportional to n squared in the worst case. The claimed result therefore identifies a sharp contrast: allowing any fixed distortion above 1 brings the required dimension below every positive power of n. It would answer Naor's sublinear-dimension question and let researchers replace finite Lp configurations with much smaller coordinate representations when bounded multiplicative error is acceptable.

Are there practical applications?

The immediate value is foundational: the result would show that finite Lp geometry admits compact approximate representations without changing its distance exponent. Such representations are relevant to studying distances among data points, but the supplied claims give no construction-time or computational guarantees. Existence of an embedding alone does not establish a practical compression procedure.

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

Subpolynomial dimension reduction in Lp

September 23, 2026 19 pages Main result formalized in Lean

For every fixed $1\lt p\lt \infty$ and D > 1, every n-point subset of a real L_p space embeds into $\ell_p^d$ with distortion at most D and subpolynomial dimension $d=n^{o(1)}$ . The target has the same exponent p, and the embedding need not be linear.

PDF LaTeX source

Cite (BibTeX)

@misc{OAI:Subpolynomial-dimension-reduction-in-Lp-September-23-2026,
  author = {{OpenAI}},
  title = {{Subpolynomial dimension reduction in $L_p$}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Subpolynomial-dimension-reduction-in-Lp-September-23-2026/paper.pdf}{OAI:Subpolynomial-dimension-reduction-in-Lp-September-23-2026}},
  year = {2026}
}

Lean formalization

OpenAI's note on what the formalization covers, from lean/docs/094.md.

Subpolynomial dimension reduction in L_p

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

Subpolynomial dimension reduction in $L_p$

Scope

For $p>1$ , $p\ne2$ , the formalization bounds the least dimension needed to embed every $n$ -point subset of real $L_p$ with distortion $D$ . For each fixed $D>1$ , there is a constant $C=C(p,D)$ such that this dimension lies between $\log n/\log(1+2D)$ and $\exp(C(\log n)^{\gamma(p)})$ for every $n\ge2$ , where $\gamma(p)=2-p$ for $p<2$ and $1-2/p$ for $p>2$ . At distortion $1$ and $n\ge9$ , it lies between $\lfloor(n-1)/4\rfloor^2$ and $\binom n2$ . The least dimension is attained, and its logarithm divided by $\log n$ tends to $0$ for $D>1$ and to $2$ for $D=1$ .

Comparator links

Result	Comparator statement
Dimension reduction in $L_p$	SubpolynomialLp.lean

At a glance

Subject: Convex and metric geometry
Manuscripts: 1
Pages: 19
Dated: Sep 23, 2026
Lean formalization: Yes
Reasoning summary: No

Named problems addressed

Naor's sublinear-dimension question for $p \ne 2$ Proved

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

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On the map

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