[#P2896] Logarithmic dimension for almost-equilateral sets in Banach spaces
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 \(i,j\)?
1Context
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.
2Definitions
Definition 1. A finite-dimensional real Banach space is a finite-dimensional real vector space equipped with a norm; completeness is automatic in finite dimension.
Definition 2. An epsilon-almost-equilateral set at scale 1 is a set whose pairwise distances all lie in [1-epsilon,1+epsilon].
3What 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.
1Status
Current status (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.[2][4][3][1]
1Records
Notes and companion material
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.
- Independent source, duplicate, exact-title, and equivalent-formulation review completed on 2026-08-01.
Recorded example 1. Finite-dimensional Hilbert spaces satisfy the conclusion by standard concentration, and the cited coding-theoretic construction handles every classical l_p^d family.
2See also
- An infinite-dimensional Banach space where every operator attains its normbanach spaces
- Conway’s thrackle conjecturediscrete geometry
- Borsuk’s conjecture in four dimensionsdiscrete geometry
How to cite
TheoremDB contributors, “Logarithmic dimension for almost-equilateral sets in Banach spaces,” TheoremDB research memory, snapshot of August 1, 2026. https://theoremdb.org/statements/almost-equilateral-banach-logarithmic-dimensionThis page as plain text: almost-equilateral-banach-logarithmic-dimension.md
This problem includes 2 records joined by 2 typed links, sourced from mathoverflow.net[1], current as of August 1, 2026.
1References
- Packet source. MathOverflow question 188161, “Logarithmic dimension for almost-equilateral sets in Banach spaces,” checked 2026-08-01. Question statement, visible answers and comments, or the linked article sections described in the source record. ↗forum · reference source · checked 2026-08-01Source use: original summary.Supports the exact formulation, the nearest published result, or the unresolved boundary recorded for this problem.Also cited at Question 188161 and all visible comments, checked through the Stack Exchange API on 2026-07-27.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.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.
- 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. Question statement, visible answers and comments, or the linked article sections described in the source record. ↗journal article · primary source · checked 2026-08-01Source use: original summary.Supports the exact formulation, the nearest published result, or the unresolved boundary recorded for this problem.Also cited at main low-dimensional embedding theorem and coding construction for finite ultrametrics in classical l_p spaces.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.
- 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. Question statement, visible answers and comments, or the linked article sections described in the source record. ↗journal article · primary source · checked 2026-08-01Source use: original summary.Supports the exact formulation, the nearest published result, or the unresolved boundary recorded for this problem.Also cited at abstract and main theorem on distance concentration in a normed-space unit ball.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.
- arXiv preprint 0406264, linked primary source for “Logarithmic dimension for almost-equilateral sets in Banach spaces,” checked 2026-08-01. Question statement, visible answers and comments, or the linked article sections described in the source record. ↗preprint · primary source · arXiv:math/0406264, checked 2026-08-01 · checked 2026-08-01Source use: original summary.Supports the exact formulation, the nearest published result, or the unresolved boundary recorded for this problem.Also cited at Introduction, which expressly places almost-equilateral sets outside the paper's scope; Section 9.2 gives references and Problem 7.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.
This is an original CC0 textbook restatement motivated by the cited MathOverflow thread; no MathOverflow prose was copied.