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Maximal Projection Constants and Extremal Vector Configurations: Some Conjectures and Examples

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Pith's one-line read The paper conjectures that the maximal absolute projection constant equals 16/7 in dimension 6 and 8/3 in dimension 8, attained by the edge midpoints of a regular simplex, and proves a general lower bound of 4(m−2)/(m+1) for every m≥3.

desk verdict A clean examples-and-conjectures note: the proven lower bound via edge-midpoint frames is solid, but the headline exact values in dimensions 6 and 8 rest on undocumented numerical searches and should be treated as well-motivated guesses, not results. read the letter →

arxiv 2608.08695 v1 pith:BYHHMBD3 submitted 2026-08-09 math.FA

Beata Deregowska , Barbara Lewandowska This is my paper · ORCID
classification math.FA MSC 46B2015A4242C15
keywords maximalabsoluteprojectionconstantrelativequasimaximaltightframesbiangularspherical(22)-designsSeidelmatricesedgemidpointsofregularsimplex
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper tries to pin down the largest possible absolute projection constant λ_R(m) among m-dimensional real Banach spaces, a number known exactly in only a few dimensions. Its concrete proposal is that in dimensions 6 and 8 the extremal configuration is the set of midpoints of the edges of a regular simplex, giving λ_R(6)=16/7 and λ_R(8)=8/3. The supporting result is a general lower bound: for every m≥3 these edge midpoints form a biangular Parseval tight frame whose relative and quasimaximal projection constants both equal $4(m-2)/(m+1)$. The same analysis, together with examples in dimensions 4 and 5, leads the authors to conjecture that weighted spherical (2,2)-designs—and ultimately only the sign pattern of their Gram matrices—determine maximal projection constants. If true, the problem acquires a finite, combinatorial face that existing methods can attack.

What carries the argument

The mechanism carrying the argument is the identity $\lambda_{\mathbb K}(U)=\mu_{\mathbb K}(U)$ for biangular tight frames. A biangular tight frame is a Parseval frame whose pairwise inner-product magnitudes take exactly two values; the equidistribution property fixes the number of large entries in each row, and a Cauchy–Schwarz style row-sum argument shows that the uniform weight vector already maximizes the quadratic form defining $\lambda_{\mathbb K}(U)$. The edge-midpoint configuration $V_m$ enters as an explicit biangular Parseval frame with exactly $m(m+1)/2$ vectors, and computing its row sums gives the value $4(m-2)/(m+1)$. The second mechanism is the spherical-design inequality of Theorem 3.1, which yields the König–Tomczak-Jaegermann upper bounds, together with the sign-matrix reformulation via Seidel matrices, which turns the maximization into one over finite combinatorial sign patterns.

What would settle it

Run a global-optimization search over all Parseval frames $U\in\mathbb R^{6\times 15}$ and $U\in\mathbb R^{8\times 36}$, starting from the explicit edge-midpoint frames; if any such frame has $\lambda_{\mathbb R}(U)>16/7$ or $\lambda_{\mathbb R}(U)>8/3$ respectively, Conjecture 2.1 is false.

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Extended reading notes

Core claim

The central discovery is that for biangular tight frames the relative projection constant and the quasimaximal relative projection constant coincide, because the equidistribution property makes every row of the Gram matrix contribute the same weighted sum. Applying this to the $m(m+1)/2$ midpoints of the edges of a regular $m$-simplex, which form a biangular Parseval frame whose inner products take exactly two absolute values, gives $\lambda_{\mathbb R}(V_m)=4(m-2)/(m+1)$ and hence the lower bound $\lambda_{\mathbb R}(m)\ge 4(m-2)/(m+1)$. For $m=6$ and $m=8$ the gap between this value and the best known upper bound is small, and the paper's numerical evidence agrees with equality; the paper therefore conjectures $\lambda_{\mathbb R}(6)=16/7$ and $\lambda_{\mathbb R}(8)=8/3$. A separate thread identifies weighted spherical (2,2)-designs with the smallest possible number of vectors as candidate extremizers in dimensions 4 and 5, where the relevant quantity appears to depend only on the sign matrix of the Gram matrix rather than on the exact inner products.

Load-bearing premise

The conjectured equalities in dimensions 6 and 8 rest on the unproven premise that the numerical searches for $\lambda_{\mathbb R}(6)$ and $\lambda_{\mathbb R}(8)$ found a global maximum, not merely a local one; if some other frame exceeded $16/7$ or $8/3$, the conjectures would fail.

Editorial extensions

If this is right

  • If Conjecture 2.1 is correct, $\lambda_{\mathbb R}(6)=16/7$ and $\lambda_{\mathbb R}(8)=8/3$ become the first exact maximal absolute projection constants in these two dimensions.
  • The lower bound $\lambda_{\mathbb R}(m)\ge 4(m-2)/(m+1)$ holds in every dimension $m\ge 3$, so the edge-midpoint configuration is always a certified starting point for searching exact values.
  • The equality of relative and quasimaximal constants for biangular tight frames means the uniform weighting is optimal for such frames, simplifying numerical tests of extremality.
  • If Conjecture 3.1 is correct, then for weighted spherical (2,2)-designs with no orthogonal pair the maximal projection constant can be obtained by maximizing only over Seidel sign matrices, a finite combinatorial problem for each fixed $N$.
  • The examples in dimensions 4 and 5 suggest that sign matrices, rather than exact inner products, may be the object that governs extremal projection constants.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The proof of Lemma 2.1 is field-independent, so the same equality should hold for complex biangular tight frames; if a complex analogue of the edge-midpoint frame exists, it would yield analogous lower bounds for $\lambda_{\mathbb C}(m)$.
  • The pattern in dimensions 6 and 8 suggests a heuristic: when a real maximal equiangular tight frame is impossible, the extremizer may be the next most symmetric two-distance tight frame with $N=m(m+1)/2$ vectors, singling out the edge-midpoint family as a systematic search space.
  • If Conjecture 3.1 transfers to arbitrary $N$, then $\lambda_{\mathbb R}(m)$ could be computed by enumerating Seidel matrices up to a bound related to minimal spherical designs, connecting projection constants to spectral graph theory in a concrete way.
  • A natural higher-precision test would be a random-restart optimization over the Stiefel manifold of Parseval frames for $m=6$ and $m=8$; if many independent runs plateau at $16/7$ or $8/3$ and never exceed them, that would strengthen, though not prove, the conjecture.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 6 minor

Summary. This paper studies the maximal absolute projection constant λ_K(m) and candidate extremal configurations. After recalling the frame-theoretic formulations (2) and (9), the authors prove Lemma 2.1, stating that for any biangular Parseval tight frame the relative and quasimaximal constants coincide. They then construct, in Example 2.1, the frame V_m formed by the midpoints of edges of a regular m-simplex, verify that it is a biangular Parseval tight frame, and compute λ_R(V_m)=4(m-2)/(m+1), yielding Corollary 2.1. They conjecture that this lower bound is sharp for m=6 and m=8. Section 3 gives a proof of the known König–Tomczak-Jaegermann upper bound via the spherical (2,2)-design inequality, and discusses weighted spherical (2,2)-designs in R^4 and R^5 whose sign matrices coincide with those of conjectured maximizers, leading to Conjecture 3.1.

Significance. The paper's main proven contribution, Lemma 2.1 and the edge-midpoint frame in Example 2.1, is explicit and verifiable; the computation gives a clean lower bound for λ_R(m). The conjectures are clearly stated and are falsifiable, and the connection between biangular tight frames and quasimaximal constants is a useful structural observation. The proof of Theorem 3.2, once the omitted algebra is supplied, provides a compact route to a known bound. The numerical evidence for Conjecture 2.1 is not documented in detail, so the conjectures should be read as plausible but not strongly established. Overall this is a useful contribution to the study of extremal projection constants, though its main claims are conjectural.

major comments (2)
  1. [§3, Theorem 3.2, Eq. (8)] The step described as 'Squaring the addends of the left-sided sum and rearranging the latter inequality' is a substantial algebraic manipulation. Let A=Σ t_i t_j |⟨u_i,u_j⟩|²/(‖u_i‖‖u_j‖), B=Σ t_i t_j |⟨u_i,u_j⟩|, and C=Σ t_i t_j ‖u_i‖‖u_j‖. The displayed inequality reads A-2φB+φ²C ≥ (c₂C-2φ²A+φ⁴C)/(1+φ)², and rearranging yields (8). Please write out these intermediate steps; in the current form the proof of Theorem 3.2 cannot be checked without reconstructing this algebra.
  2. [§2, Conjecture 2.1] The conjecture that λ_R(6)=16/7 and λ_R(8)=8/3 rests on the small gap between λ_R(V_m) and the non-sharp upper bound of Theorem 1.3, and on numerical estimates attributed to Chalmers and to [25]. The numerical searches behind (2) and (9) are not described, and the optimization problems are non-concave, so without details about the algorithms, tolerances, or optimality certificates the reader cannot assess whether these are global maxima. Please either document the numerical evidence or explicitly label Conjecture 2.1 as heuristic.
minor comments (6)
  1. [§3, Theorem 3.2 proof] The reduction to nonzero columns is too terse: after deleting zero columns, the coefficient vector t should be restricted and renormalized; the displayed inequality with the same t is only valid after this is explained.
  2. [§3, after Theorem 3.4] The sentence 'A = sgn(U^T U) realizes the maximum in (9)' appears inconsistent with the definition A=I_N+B in (9), since the latter has off-diagonal entries in {0,2}. Please state precisely how the sign matrix is embedded in the Seidel-matrix formulation.
  3. [Conjecture 3.1] The notation 'U ∈ R^m' is incorrect for a sequence of vectors; also the definition of sgn(U^T U) for vanishing inner products should be stated explicitly.
  4. [Example 2.1] The combinatorial counts in the displayed sum (how many pairs fall into each of the three categories) would be easier to verify if a short counting argument were included.
  5. [References] Reference [16] has a typo: 'Small number of vectors,,' should be 'small number of vectors'.
  6. [Figure 1] The figure needs axis labels, a caption, and a non-color legend; the current 'yellow dots' vs 'blue ones' description is hard to use in print.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the proven bounds are self-contained, and the conjectures are explicitly labeled as resting on external numerical searches.

full rationale

The derivation chain in this paper is self-contained where it proves results. Lemma 2.1 establishes equality of the relative and quasimaximal projection constants for biangular tight frames using the external equidistribution result [7] plus a direct inequality; the reverse inequality is immediate from the definitions. Example 2.1 verifies by direct computation that the edge-midpoint frame is biangular Parseval, and Corollary 2.1 follows from the definition of lambda_R(m) as a supremum, not from any fitted parameter. The upper bound of Theorem 1.3 is attributed to the authors' previous paper [11], but it is independently re-proved in this paper in Theorem 3.2 using the external design inequality [16], so the self-citation is not load-bearing. Conjecture 2.1 is explicitly a conjecture supported by external numerical estimates by Chalmers and by [25]; it is not derived from those searches as a theorem, and no fitted input is relabeled as a prediction. The same holds for the spherical-design examples in Section 3, which are presented as evidence and conjectures. No uniqueness theorem from the authors is invoked to force a choice, and no known result is merely renamed as a new contribution. The only weakness, the unverified globality of the numerical searches, is a correctness risk for the conjecture, not a circularity in the paper's reasoning.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The proven results lean on two external structural theorems, the equidistribution of biangular tight frames and a Welch-type design inequality, and on the authors' earlier proof of the upper bound. The conjectures lean on unpublished numerical searches. No free parameters or invented entities are used; the proof parameter phi is chosen algebraically, not fitted to data.

assumptions (5)
  • domain assumption Every biangular tight frame is equidistributed: each row of |U^*U| has the same number of b-entries and c-entries.
    Imported from [7, Proposition 5.1] and used in the proof of Lemma 2.1; no proof is reproduced.
  • domain assumption The frame optimization formula lambda_K(m,N) equals max over Parseval frames of the weighted average of |U^*U| entries, as stated in Theorem 1.1 from [5].
    Used to connect projection constants to Parseval frames throughout the paper.
  • domain assumption The weighted spherical design inequality of Theorem 3.1 from [16] holds for all unit vector configurations and nonnegative weights.
    Used in the proof of Theorem 3.2 to lower-bound the quartic sum over normalized frame elements.
  • domain assumption The size bounds for weighted spherical designs in Theorem 3.3 from [27] are correct.
    Used in Section 3 to identify the smallest possible number of vectors for weighted spherical (2,2)-designs.
  • ad hoc to paper The numerical searches cited from Chalmers and [25] are globally exhaustive for lambda_R(6) and lambda_R(8).
    Conjecture 2.1 and the dimensional examples in Section 3 depend on the reliability of these computational searches, which are not described in enough detail to verify.

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Pith. "Pith review of Maximal Projection Constants and Extremal Vector Configurations: Some Conjectures and Examples." pith.science (2026). https://pith.science/paper/BYHHMBD3

@misc{pith2026260808695,
  author       = {Pith},
  title        = {Pith review of: Maximal Projection Constants and Extremal Vector Configurations: Some Conjectures and Examples},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/BYHHMBD3}},
  note         = {Machine review of arXiv:2608.08695}
}
abstract

Let $\lambda_{\mathbb K}(m)$ denote the maximal absolute projection constant among $m$-dimensional Banach spaces over $\mathbb K=\mathbb R$ or $\mathbb C$. Its exact value is known only in a few cases, and determining it remains a challenging problem. In this note, we investigate several structured vector configurations that naturally arise in this context. We first recall the connection between maximal projection constants and tight frames, then consider biangular tight frames, and show that their relative and quasimaximal projection constants coincide. A particularly interesting example is provided by the midpoints of the edges of a regular simplex, which yield natural lower bounds for $\lambda_{\mathbb R}(m)$. Numerical evidence suggests that these bounds may be sharp in dimensions $6$ and $8$. We also discuss weighted spherical $(2,2)$-designs with a small number of vectors and their connection with maximal projection constants. Examples in dimensions $4$ and $5$ indicate that the sign patterns of their Gram matrices may play an important role. These observations lead us to formulate several conjectures concerning maximal absolute projection constants and the vector configurations associated with them.

Figures

Figures reproduced from arXiv: 2608.08695 by the authors.

Figure 1
Figure 1. Yellow dots show upper bound for λR(m), blue ones show value of λR(Vm) For m = 7, these vectors form the maximal equiangular tight frame and λR(V7) = λR(7). Moreover, as illustrated in [PITH_FULL_IMAGE:figures/full_fig_p009_1.png] view at source ↗

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