REVIEW 2 major objections 4 minor 19 references
Spaces with the maximal projection constant revisited
T0 review · 2 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash
Pith's one-line read For dimensions with a maximal equiangular tight frame, the extremal projection-constant spaces are exactly those whose dual unit ball is sandwiched between the frame's absolutely convex hull and a rescaled zonotope.
desk verdict Solid characterization, but the proof that R^2 is the only unique dimension has a genuine gap in the C^3 case. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
The load-bearing object is the maximal equiangular tight frame (ETF): a set of $d_{\mathbb{K}}(n)$ unit vectors in $\mathbb{K}^n$ for which $|\langle w_i,w_j\rangle|$ is constant for $i\neq j$, where $d_{\mathbb{R}}(n)=n(n+1)/2$ and $d_{\mathbb{C}}(n)=n^2$. This frame does double work: it is the only configuration that can attain equality in the estimate $\lambda_{\mathbb{K}}(n)\le\delta_{\mathbb{K}}(n)$, and it generates both sides of the characterizing inclusion, the absolutely convex hull $\operatorname{absconv}\{w_1,\dots,w_d\}$ and the rescaled zonotope $\frac{n}{d\,\delta_{\mathbb{K}}(n)}Z(w_1,\dots,w_d)$. The mechanism that carries the argument is the Chalmers–Metcalf operator $E$ from Lemma 3.4, a linear operator on $\ell_\infty^N$ whose trace on the embedded subspace equals the projection constant; Theorem 1.1's equality conditions force $E$ to act as $\frac{\delta_{\mathbb{K}}(n)}{n}\operatorname{Id}$ on the subspace, and the rows of $E$ outside the frame's support contribute exactly the extra points of the zonotope appearing in $B_{X^*}$.
What would settle it
In $\mathbb{R}^2$, take any centrally symmetric convex hexagon that is not affinely regular, realize the corresponding norm, and compute its absolute projection constant by solving the minimal-projection problem for its isometric embedding into $\ell_\infty^6$. If the result equals $4/3$, Corollary 3.6's uniqueness claim is wrong; finding a value strictly below $4/3$ supports the paper's assertion that only the affine regular hexagon is extremal.
Extended reading notes
Core claim
The central claim is Theorem 1.2. If $\mathbb{K}^n$ contains a maximal equiangular tight frame $w_1,\dots,w_d$ with $d=d_{\mathbb{K}}(n)$, then an $n$-dimensional normed space $X=(\mathbb{K}^n,\|\cdot\|)$ satisfies $\lambda(X)=\lambda_{\mathbb{K}}(n)$ if and only if there is a linear transformation $T$ such that $\operatorname{absconv}\{w_1,\dots,w_d\}\subseteq T(B_{X^*})\subseteq \frac{n}{d\,\delta_{\mathbb{K}}(n)}Z(w_1,\dots,w_d)$, where $Z(w_1,\dots,w_d)$ is the Minkowski sum of the segments $\operatorname{absconv}\{w_j\}$ and is a zonotope in the real case. The proof rests on a sharpened version of the trace-duality estimate $\lambda_{\mathbb{K}}(n)\le \delta_{\mathbb{K}}(n)$: Theorem 1.1 gives the full equality conditions, forcing every nonzero frame vector in an extremal configuration to lie along a fixed maximal ETF, with norm $\sqrt{n}\,t_i$ and with squared weights $1/d$ distributed evenly over the $d$ directions. The equality conditions then force the associated Chalmers–Metcalf projection operator to be a scalar multiple of the identity on the embedded extremal subspace, which is exactly what yields the sandwich. The same mechanism shows that the two bounding polytopes coincide only when $n=2$ and $\mathbb{K}=\mathbb{R}$, producing the affine regular hexagon, and otherwise the extremal norms form an infinite family.
Load-bearing premise
The paper only treats dimensions in which an equiangular tight frame of the maximal possible size $d_{\mathbb{K}}(n)$ actually exists in $\mathbb{K}^n$; in dimensions without such a frame, the value $\lambda_{\mathbb{K}}(n)$ itself is not known, so the characterization has no subject.
Editorial extensions
If this is right
- In every covered dimension, the maximal absolute projection constant equals $\delta_{\mathbb{K}}(n)$, and every extremal space has its dual unit ball pinned between the two explicitly constructed frame polytopes.
- Outside the real plane, there are infinitely many pairwise non-isometric $n$-dimensional spaces with the maximal projection constant, so uniqueness is a phenomenon of $\mathbb{R}^2$ alone.
- Equality in the trace-duality estimate forces the underlying configuration to be a maximal ETF with evenly distributed weights, so any numerical optimizer that reaches $\delta_{\mathbb{K}}(n)$ automatically produces a maximal ETF.
- For quasimaximal relative projection constants with equal weights, the value $\mu_{\mathbb{K}}(n,N)$ equals $\lambda_{\mathbb{K}}(n)$ exactly when $N$ is a multiple of $d_{\mathbb{K}}(n)$.
- In every covered dimension other than $\mathbb{R}^2$ there are non-polytopal extremal norms, because the zonotope part of the sandwich can be chosen with infinitely many independent generator points.
Reading between the lines
- If the Zauner conjecture is eventually proved, a SIC-POVM in every complex dimension is exactly a maximal ETF, and Theorem 1.2 would then cover all complex dimensions at once; until then its reach is limited to the finitely many known maximal-ETF dimensions.
- The two bounding polytopes in the sandwich define an interval of convex bodies, so the family of extremal norms in a fixed covered dimension carries a natural convex-geometric parameterization; one could try to compute its Banach–Mazur diameter as a function of $n$.
- A testable computational extension is to take the known one-parameter family of maximal ETFs in $\mathbb{C}^3$, build the sandwiched dual balls for a grid of parameter values, and verify numerically that each resulting norm has projection constant $\delta_{\mathbb{C}}(3)$.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper studies n-dimensional normed spaces over K=R or K=C with maximal absolute projection constant λ_K(n). In dimensions where a maximal equiangular tight frame (ETF) exists, it proposes a complete characterization: X has λ(X)=λ_K(n) iff, after a linear change of coordinates, the dual unit ball is sandwiched between absconv of the ETF vectors and a rescaled zonotope generated by the same vectors (Theorem 1.2). It also gives equality conditions in the trace-duality estimate that underlies the bound λ_K(n)≤δ_K(n) (Theorem 1.1), and it derives a uniqueness dichotomy: up to isometry, the only unique maximizer is the real two-dimensional case, whose unit ball is an affine regular hexagon (Corollary 3.6). The paper contains a detailed historical account of earlier incorrect proofs and builds on the recent simple proof of the Grünbaum conjecture.
Significance. If the characterization and uniqueness dichotomy are correct, the paper resolves a long-standing question for all dimensions where a maximal ETF is known to exist, including R^2, R^3, R^7, R^23 and many complex dimensions. The equality conditions in Theorem 1.1 are substantially more precise than previous results and are needed for the sandwich characterization. The paper also correctly highlights that the dimensions covered are exactly those where the value of λ_K(n) is currently known. The main theorems are plausible, but two load-bearing steps in the proofs as written need repair before the claims can be accepted.
major comments (2)
- [Lemma 3.2, case n=3, K=C] The strict-inclusion proof for C^3 contains an invalid inference. After defining S={i≥2:⟨w_i,w_1⟩/⟨w_i,v⟩=φ/M}, the text claims that from M≥|⟨w_i,v⟩| and Re⟨w_i,v⟩<M it follows that Re(φ/⟨w_i,v⟩)>φ/M. This is false: for example z=iM satisfies |z|=M and Re z<M, yet Re(φ/z)=0<φ/M. Consequently the asserted negativity of the left-hand derivatives at 1/(2M) for all i∉S is not established, and the contradiction proving strictness of the inclusion (1.2) does not follow. Since Corollary 3.6 uses strictness of this inclusion to rule out uniqueness in C^3, the uniqueness dichotomy is not proved as written, although Theorem 1.2 itself may be unaffected.
- [Proof of Theorem 1.2, final containment in the zonotope] In the forward direction, after deriving the representation for the j-th coordinate functional with j∉S, the proof defines w=1/(dC)Σ_{s=1}^d a_s w_s with a_s=dα_s and |a_s|≤1, and asserts that w∈ n/(dδ)Z(w_1,...,w_d). This containment requires |a_s|≤dC^2, because the coefficients of w are a_s/(dC) and the allowed coefficient bound in n/(dδ)Z is C=n/(dδ). The established bound |a_s|≤1 is insufficient; already for K=R, n=3 one has dC^2≈0.573<1. The missing sharper estimate on α_s=Σ_{k∈A_s}E(e_k)_j is exactly what is needed to prove T(B_{X*})⊆n/(dδ)Z, so the forward implication of Theorem 1.2 is incomplete as written.
minor comments (4)
- [Equations (3.5)–(3.6) and surrounding text] Several displayed sums in the proof of Theorem 1.2, such as Σ_i x_{ik}x_{ij}, omit complex conjugates. With the standard inner product on K^n, the identification of this sum with ⟨u_j,u_k⟩ requires conjugating one factor. Please clarify the inner-product convention and insert the missing conjugates, since the complex case is part of the main theorem.
- [Lemma 3.2, R^2 equality part] The step 'Without loss of generality we can assume that a_1≥a_2≥a_3' needs justification: permuting the coefficients changes the vector unless one first uses the relation w_1+w_2+w_3=0 to shift all coefficients by the same constant. Adding such a constant preserves the point and allows ordering the coefficients while keeping them in [0,1/2] up to absolute value.
- [Corollary 3.6] The assertion that K_i cannot be generated by fewer than d+i points is stated without proof. The construction only shows that x_i∉K_{i-1}; to conclude that the minimal generator count is exactly d+i, one should verify that each x_i is an extreme point of K_i, or give a separate argument for the linear-invariance claim.
- [Abstract and Introduction] There are minor typographical issues, e.g. 'to the the general Gerzon upper bound' in the abstract and a stray period in the display of δ_K(n). These do not affect the mathematics.
Circularity Check
No circularity: the main theorems are proved from scratch and rely only on external, independently established results.
full rationale
The paper's derivation chain is self-contained rather than circular. Theorem 1.1 proves the upper bound and, importantly, the equality conditions for the estimate λ_K(n) ≤ δ_K(n) directly from the Cauchy-Schwarz inequality, trace inequalities, and frame identities; it does not assume the target characterization. Theorem 1.2 then uses those equality conditions together with Lemma 3.4, which is explicitly an external lemma from König and Tomczak-Jaegermann, to derive the sandwich characterization of maximal projection constant spaces. Lemma 3.1 and Lemma 3.3 are algebraic consequences of ETF properties and of Theorem 1.1's equality conditions, not restatements of the conclusion. The paper's conditional assumption that a maximal ETF exists is stated as an external hypothesis, and the characterization is not defined in terms of the target equality. The known estimate λ_K(n) ≤ δ_K(n) is used as a starting point, but the equality analysis and the isometric description of all extremal spaces are new content proved in the paper. There are no self-citations by the present author that carry the argument, and the cited results are independent prior theorems rather than claims unique to this paper, so no self-citation chain is load-bearing. The reviewer's concern about the strict-inclusion argument in Lemma 3.2 is a potential correctness defect in one step of Corollary 3.6; it is not a circularity, because the claimed inclusion is not used as its own premise and Theorem 1.2 is stated not to depend on the strictness part. Accordingly, the correct circularity finding is no significant circularity.
Assumptions & free parameters
assumptions (5)
- domain assumption The existence of a maximal equiangular tight frame in K^n for the considered n (i.e., d_K(n) vectors).
- standard math The upper bound λ_K(n) ≤ δ_K(n), proved by Deręgowska and Lewandowska using the Bukh-Cox result.
- standard math Kadec-Snobar theorem: λ(X) < √n for dim X = n.
- standard math The fact that any maximal equiangular set of vectors is automatically a tight frame (see Theorem 5.10 in Foucart-Rauhut).
- standard math Lemma 3.5 (algebraic lemma about traces on quotient spaces) is stated without proof.
Cite this review
Pith. "Pith review of Spaces with the maximal projection constant revisited." pith.science (2026). https://pith.science/paper/SL2YLAAL
@misc{pith2026250524526,
author = {Pith},
title = {Pith review of: Spaces with the maximal projection constant revisited},
year = {2026},
howpublished = {\url{https://pith.science/paper/SL2YLAAL}},
note = {Machine review of arXiv:2505.24526}
}
abstract
Let $n \geq 2$ be an integer such that an equiangular set of vectors $w_1, \ldots, w_d$ of the maximal possible cardinality (in relation to the the general Gerzon upper bound) exists in $\mathbb{K}^n$, where $\mathbb{K}=\mathbb{R}$ or $\mathbb{K}=\mathbb{C}$ (i.e. $d=\frac{n(n+1)}{2}$ in the real and $d=n^2$ in the complex case). We provide a complete characterization of $n$-dimensional normed spaces $X$ having a maximal absolute projection constant among all $n$-dimensional normed spaces over $\mathbb{K}$. The characterization states that $X$ has a maximal projection constant if and only if it is isometric to a space, for which the unit ball of the dual space is contained between the absolutely convex hull of the vectors $w_1, \ldots, w_d$ and an appropriately rescaled zonotope generated by the same vectors. As a consequence, we obtain that in the considered situations, the case of $n=2$ and $\mathbb{K}=\mathbb{R}$ is the only one, where there is a unique norm in $\mathbb{K}^n$ (up to an isometry) with the maximal projection constant. In this case, the unit ball is an affine regular hexagon in $\mathbb{R}^2$.
Reference graph
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Reviewed August 7, 2026 · model on record in the stance chip above.
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