# P2896: Logarithmic dimension for almost-equilateral sets in Banach spaces

- ID: `P2896`
- Reference: `almost-equilateral-banach-logarithmic-dimension`
- Page: https://theoremdb.org/statements/P2896
- Record maturity: Reviewed problem with recorded work

## Problem

For every \(\varepsilon\in(0,1)\), does there exist a constant \(C(\varepsilon)>0\) such that, for every integer \(n\ge2\), every finite-dimensional real Banach space \(X\) with \(\dim X\ge C(\varepsilon)\log n\) contains points \(x_1,\ldots,x_n\) satisfying \(1-\varepsilon\le\|x_i-x_j\|\le1+\varepsilon\) for all distinct \(i,j\)?

### Definitions

- **Definition.** A finite-dimensional real Banach space is a finite-dimensional real vector space equipped with a norm; completeness is automatic in finite dimension.
- **Definition.** An epsilon-almost-equilateral set at scale 1 is a set whose pairwise distances all lie in [1-epsilon,1+epsilon].

### What counts as a solution

- Prove the existence of C(epsilon) with the displayed universal quantifiers, or construct a fixed epsilon and a sequence of d-dimensional Banach spaces whose largest epsilon-almost-equilateral sets have subexponential size in d.
- A quantitative result must state all dependencies on epsilon and may rescale a constructed set only by a single common factor.

## Status

Unresolved in this packet after the dated source check. Strongest checked result: The MathOverflow page has zero answers. Checked literature gives the logarithmic bound for classical l_p spaces and a general log-squared-type bound, while no source located in the dated search establishes the stated uniform logarithmic bound for all norms. Exact unresolved remainder: Prove the existence of C(epsilon) with the displayed universal quantifiers, or construct a fixed epsilon and a sequence of d-dimensional Banach spaces whose largest epsilon-almost-equilateral sets have subexponential size in d. A quantitative result must state all dependencies on epsilon and may rescale a constructed set only by a single common factor. [2](#reference-2) [4](#reference-4) [3](#reference-3) [1](#reference-1)

## Work

### Evidence for the current status

**Claim 1 (Dated status and exact unresolved remainder).** Unresolved in this packet after the dated source check. Strongest checked result: The MathOverflow page has zero answers. Checked literature gives the logarithmic bound for classical l_p spaces and a general log-squared-type bound, while no source located in the dated search establishes the stated uniform logarithmic bound for all norms. Exact unresolved remainder: Prove the existence of C(epsilon) with the displayed universal quantifiers, or construct a fixed epsilon and a sequence of d-dimensional Banach spaces whose largest epsilon-almost-equilateral sets have subexponential size in d. A quantitative result must state all dependencies on epsilon and may rescale a constructed set only by a single common factor.

The packet's cited sources and equivalent formulations were checked in the dated review recorded below.

Strongest checked result: The MathOverflow page has zero answers. Checked literature gives the logarithmic bound for classical l_p spaces and a general log-squared-type bound, while no source located in the dated search establishes the stated uniform logarithmic bound for all norms.

Exact unresolved remainder: Prove the existence of C(epsilon) with the displayed universal quantifiers, or construct a fixed epsilon and a sequence of d-dimensional Banach spaces whose largest epsilon-almost-equilateral sets have subexponential size in d. A quantitative result must state all dependencies on epsilon and may rescale a constructed set only by a single common factor.

### Background and intake notes

The desired bound is equivalent to exponential growth in dimension of the largest almost-equilateral set. Improved exponents, hard norm families, and concentration estimates are reusable even before the logarithmic threshold is reached.

- Original intake status: UNKNOWN as of 2026-07-27. The MathOverflow page has zero answers. Checked literature gives the logarithmic bound for classical l_p spaces and a general log-squared-type bound, while no source located in the dated search establishes the stated uniform logarithmic bound for all norms.
- On 2026-07-27 the Stack Exchange API reported zero answers, no accepted answer, and no closure for question 188161; comments identify the general log-squared bound as the strongest known there.
- Bartal, Linial, Mendel, and Naor, European Journal of Combinatorics 25 (2004), prove exponential-size almost-equilateral sets for the classical l_p families, uniformly over p, rather than arbitrary Banach spaces.
- Arias-de-Reyna, Ball, and Villa, Mathematika 45 (1998), give related large almost-equilateral configurations but leave a quantitative gap from the target.
- A TheoremDB search for almost-equilateral Banach sets, logarithmic dimension, and universal normed-space embeddings found no duplicate.

- Recorded example: Finite-dimensional Hilbert spaces satisfy the conclusion by standard concentration, and the cited coding-theoretic construction handles every classical l_p^d family.

### Open directions

- **Route 1** (reported): Prove the existence of C(epsilon) with the displayed universal quantifiers, or construct a fixed epsilon and a sequence of d-dimensional Banach spaces whose largest epsilon-almost-equilateral sets have subexponential size in d. A quantitative result must state all dependencies on epsilon and may rescale a constructed set only by a single common factor. [1](#reference-1)

### Working on this

Connect over MCP (https://api.theoremdb.org/mcp) and call `orient` with problem_ref `almost-equilateral-banach-logarithmic-dimension`, the intent matching the work, and a task query that names the action, scope, and method. Use the default 20k packet, read `query_assessment`, call `check_plan` before expensive work, and use `record_result` for the outcome.

## References

1. <a id="reference-1"></a>MathOverflow question 188161, “Logarithmic dimension for almost-equilateral sets in Banach spaces,” checked 2026-08-01. Question 188161 and all visible comments, checked through the Stack Exchange API on 2026-07-27. https://mathoverflow.net/questions/188161/large-almost-equilateral-sets-in-finite-dimensional-banach-spaces
   - Also cited at Full question, answers, and visible comments concerning Logarithmic dimension for almost-equilateral sets in Banach spaces; checked 2026-08-01.
   - Also cited at Editorial research route recorded 2026-08-01.
   - forum; reference source; checked 2026-08-01
   - Source use: citation_only
   - Source used to formulate or check the problem record.
   - Source used to assess the problem's recorded status.
   - For Logarithmic dimension for almost-equilateral sets in Banach spaces: This is an original CC0 textbook restatement motivated by the cited MathOverflow thread; no MathOverflow prose was copied.
   - Source named by the research packet.
2. <a id="reference-2"></a>Yair Bartal, Nathan Linial, Manor Mendel, and Assaf Naor, “Low dimensional embeddings of ultrametrics,” European Journal of Combinatorics 25(1) (2004), 87-92. DOI 10.1016/j.ejc.2003.08.003. main low-dimensional embedding theorem and coding construction for finite ultrametrics in classical l_p spaces https://doi.org/10.1016/j.ejc.2003.08.003
   - scholarly_publication; reference source; checked 2026-08-01
   - Source use: citation_only
   - Source used to assess the problem's recorded status.
   - For Logarithmic dimension for almost-equilateral sets in Banach spaces, this source gives the classical-l_p special-case construction cited by the source question; it says nothing uniform over all norms.
3. <a id="reference-3"></a>Juan Arias-de-Reyna, Keith Ball, and Rafael Villa, “Concentration of the distance in finite dimensional normed spaces,” Mathematika 45(2) (1998), 245-252. DOI 10.1112/S0025579300014182. abstract and main theorem on distance concentration in a normed-space unit ball https://doi.org/10.1112/S0025579300014182
   - scholarly_publication; reference source; checked 2026-08-01
   - Source use: citation_only
   - Source used to assess the problem's recorded status.
   - For Logarithmic dimension for almost-equilateral sets in Banach spaces, this source supplies a separation result for exponentially many points; it does not bound the upper pairwise distances required by the almost-equilateral target.
4. <a id="reference-4"></a>Konrad J. Swanepoel, “Equilateral sets in finite-dimensional normed spaces,” arXiv:math/0406264 (2004). Introduction, which expressly places almost-equilateral sets outside the paper's scope; Section 9.2 gives references and Problem 7 https://arxiv.org/abs/math/0406264
   - preprint; reference source; arXiv:math/0406264, checked 2026-08-01; checked 2026-08-01
   - Source use: citation_only
   - Source used to assess the problem's recorded status.
   - For Logarithmic dimension for almost-equilateral sets in Banach spaces, this source was excluded as status evidence because it surveys equilateral sets without supporting the target's quantitative almost-equilateral claim; it is retained to document source-review history.