Extremal functions on moduli spaces and applications

Nikolaj Glazunov

REVIEW 4 major objections 5 minor 15 references

Extremal functions on moduli spaces and applications

T0 review · 4 major / 5 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read This paper gives exact critical determinants for every dyadic scaling of two-dimensional Minkowski balls, plus minimal inscribed and circumscribed hexagon areas, through a two-branch formula with crossover at p≈2.5725.

desk verdict Restates known critical determinant formulas for 2D Minkowski balls; the only new m-scaling is standard, and the claimed index-2^m sublattice interpretation is false by a factor of 2^m. read the letter →

arxiv 2411.13671 v3 pith:ZW2YNFB6 submitted 2024-11-20 math.NT

Nikolaj Glazunov This is my paper · ORCID

classification math.NT MSC 11H0611-XX52C05

keywords latticepackingcriticaldeterminantMinkowskiballdomainextremalfunctionmodulispaceinscribedhexagoncoveringconstant

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 is trying to establish that all the classical extremal quantities attached to two-dimensional Minkowski balls $D_p: |x|^p+|y|^p<1$ -- the critical determinant, the optimal lattice-packing density, and the minimal areas of inscribed and circumscribed hexagons -- are controlled by a single two-branch function on the $(p,\sigma)$-plane, and that doubling the body simply multiplies this function by $4^m$. If this is right, the hard part of Minkowski's packing problem for these curves is already contained in the known piecewise critical determinant, and every dyadic scaling follows without new optimization. The paper also states that the minimal inscribed hexagon area is exactly $3\Delta(D_p)$ and the minimal circumscribed hexagon area exactly $4\Delta(D_p)$, connecting the packing and covering sides through one six-point lattice.

What carries the argument

The load-bearing object is the Minkowski-Cohn moduli function $\Delta(p,\sigma)=(\tau+\sigma)(1+\tau^p)^{-1/p}(1+\sigma^p)^{-1/p}$ on the domain $1<p<\infty$, $1\le\sigma\le\sigma_p$. It parameterizes the determinants of admissible lattices and the areas of associated hexagons for $D_p$; minimizing over $\sigma$ gives the critical determinant, and multiplying by $4^m$ accounts for $2^m$-scaling. The second mechanism is Lemma 1's shell property: the critical lattice contains exactly three pairs of points on the Minkowski curve, $\pm a_1,\pm a_2,\pm(a_1+a_2)$, and this is what turns the determinant formula into hexagon area formulas by the factors 3 and 4.

What would settle it

Compute the critical determinant of $D_p$ numerically for several $p$ in $(2.57,2.58)$ to high precision, for example $p=2.5725$, and compare with the two branches of (22); if the minimum is attained at a value of $\sigma$ strictly between $1$ and $\sigma_p$, or if a critical lattice with more than six boundary points exists for some $p$, the piecewise formulas and the hexagon theorems fail.

Watch

Extended reading notes

Core claim

The central discovery is the exact extremal function for dyadically scaled Minkowski balls in the plane. For each $p>1$, let $\sigma_p=(2^p-1)^{1/p}$ and let $\tau_p\in[0,1)$ solve $2(1-\tau_p)^p=1+\tau_p^p$. The paper claims that for every integer $m\ge0$, the critical determinant of $2^mD_p$ is $\Delta(2^mD_p)=4^m\Delta(D_p)$, with $\Delta(D_p)$ given piecewise by $\Delta(p,1)=4^{-1/p}(1+\tau_p)/(1-\tau_p)$ for $1<p\le2$ and $p\ge p_0$, and by $\Delta(p,\sigma_p)=\sigma_p/2$ for $2\le p\le p_0$, where $p_0\approx2.5725$ is the crossover. It further claims the minimal inscribed hexagon area is $3\Delta(D_p)$ and the minimal circumscribed hexagon area is $4\Delta(D_p)$. These are presented as corollaries of the six-point critical-lattice property: the critical lattice carries exactly the boundary points $\pm a_1,\pm a_2,\pm(a_1+a_2)$.

Load-bearing premise

Everything rests on the cited theorem that the critical determinant of $D_p$ is exactly (22), and on Lemma 1, stated without proof, that every critical lattice has exactly three pairs of boundary points; if either premise fails, the formulas for scaled balls and hexagons collapse.

Editorial extensions

If this is right

For every integer $m\ge0$, the densest lattice packing of $2^mD_p$ has its density determined by $4^m$ times the two-branch value of $\Delta(D_p)$; no separate packing calculation is needed for each scale.
The minimal inscribed hexagon in $D_p$ has area $3\Delta(D_p)$, and the minimal circumscribed hexagon has area $4\Delta(D_p)$, so these two covering and packing constants are locked to the same critical lattice.
In the Euclidean case $p=2$, the formulas recover $\Delta(D_2)=\sqrt3/2$, inscribed hexagon area $3\sqrt3/2$, and circumscribed hexagon area $2\sqrt3$.
Minkowski's optimization problem for the diophantine inequality (14) is solved in two dimensions for all $p$ and all dyadic scales, conditional on the cited critical-determinant theorem.

Reading between the lines

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

The paper leaves the covering extremal function open; if a covering analogue of the six-point property holds, the minimal circumscribed hexagon found here would be a natural candidate for the covering constant's lower bound, a step the paper does not take.
Because Lemma 1 is stated without proof, the unconditional status of all hexagon results rests on a missing argument; supplying a proof of the six-point shell property would be the decisive next step.
A numerical search near $p_0\approx2.5725$ could test the branch switch: for $p$ just below $p_0$ the optimal lattice should be $\Lambda_p^{(0)}$ with determinant $\sigma_p/2$, and just above it $\Lambda_p^{(1)}$ with determinant $4^{-1/p}(1+\tau_p)/(1-\tau_p)$; the paper gives no transition analysis.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Quick take: this paper does not give a new result. The core formulas for Δ(D_p) and the hexagon areas are quoted from the author's earlier papers [13,14], and the only new-looking step, the m-scaling, is the standard identity Δ(2^m K)=4^m Δ(K). Worse, the closing sentence of Theorem 1—that the determinants in (23)–(24) are those of index-2^m sublattices of the critical lattices—is false as stated. An index-2^m sublattice would have determinant 2^m Δ(D_p), while the formulas are 4^m Δ(D_p). The scaled lattice 2^m Λ is a superlattice of index 4^m, not a sublattice of index 2^m. For p=2, m=1, an index-2 sublattice of the hexagonal lattice has determinant √3 and still contains the unit vector (1,0), so it is not even admissible for 2D_2. So the sublattice interpretation is internally inconsistent with the displayed formulas.

What the paper does do is collect a set of known results—Cohn's, Watson's, Davis's, and the author's own work on Minkowski's conjecture—into a 'moduli space' framework. That could be useful for someone wanting a unified notation for packing constants and hexagon areas. The formulas themselves, including the piecewise crossover at p0≈2.5725, match the known values. The paper is honest that the covering extremal function remains open.

The soft spots, in addition to the index error: no proofs are supplied for Theorems 1–3; they are simply delegated to [13] and [14]. Lemma 1, asserting that the critical lattice always has exactly six boundary points, is unproved and is load-bearing for the hexagon theorems. The novelty is mostly notational, and the reliance on self-citations without independent verification is heavy.

If you are working on geometry of numbers or lattice packing, you already know these results from the original sources; the moduli-space terminology does not change the content. A reader new to the area might get a useful survey-style overview, but they should not cite it as a primary source.

Recommendation: I would not send this to a serious referee as is. The index error is a concrete false statement, and without proofs the rest is a restatement of earlier work. If the author fixes the index statement and provides a proof of Lemma 1 or a reference with one, then it could be a useful note, but at this point I'd recommend a desk reject.

Referee Report

4 major / 5 minor

Summary. The paper studies extremal functions attached to two-dimensional Minkowski balls D_p = {|x|^p + |y|^p < 1} and their scalings 2^m D_p. It claims three main theorems: Theorem 1 gives the critical determinants of 2^m D_p in the piecewise form (23)-(24); Theorem 2 gives the minimal area of inscribed hexagons (26); Theorem 3 gives the minimal area of circumscribed hexagons (27). The derivations rest on a cited result from the author's earlier work [13] for the critical determinant of D_p, formula (22), and on Lemma 1 asserting that the critical lattice always contains exactly three pairs of boundary points. The paper also introduces a moduli-space formalism connecting these quantities to Diophantine approximation, packing, and covering problems.

Significance. If the main claims were fully proved, the paper would offer a compact unified description of packing constants and optimal hexagons for l^p balls. The numerical content is correct in known special cases: for p = 2 the formulas reproduce the classical constants 3√3/2 and 2√3, and formulas (23)-(24) are exactly the standard scaling 4^m Δ(D_p) of the critical determinant from [13]. The paper also clearly organizes several classical results and states a genuinely interesting question about covering constants in §6.3. However, the paper does not supply proofs of its new theorems; Theorem 1 is a scaling of [13], and Theorems 2 and 3 are formal corollaries of the same input plus the unproved six-point lemma. The contribution is therefore mostly organizational, and the load-bearing Lemmas and endpoint-minimization steps are not justified in this manuscript.

major comments (4)

[§6.2, Theorem 1 (end)] The sentence "these are the determinants of the sublattices of index 2^m of the critical lattices of the corresponding balls D_p" is inconsistent with (23)-(24). A sublattice of index 2^m has determinant 2^m Δ(D_p), whereas formulas (23)-(24) give 4^m Δ(D_p). The lattice with the latter determinant is the scaled lattice 2^m Λ, which contains Λ as a sublattice of index 4^m, not as a sublattice of index 2^m. The concrete p = 2, m = 1 case makes the failure visible: Δ(D_2) = √3/2, so formula (23) gives 2√3, but no index-2 sublattice of the hexagonal lattice has determinant 2√3, and every index-2 sublattice contains a nonzero vector of length at most 2 by Minkowski's convex body theorem, so it is not admissible for the radius-2 disk. The numerical values in (23)-(24) are correct as the standard scaling 4^m Δ(D_p), but the sublattice interpretation must be corrected.
[§3.1, Lemma 1] Lemma 1 asserts that for any point (Px, Py) of a critical lattice on the Minkowski curve, the point (u, v) solving (10) lies in the lattice and on the curve, and that the shell contains exactly six points. No proof is given and no reference is cited for this lemma. This property is load-bearing: it is used in §4 to reduce the optimization problem to lattices with three pairs of boundary points, and it underlies Theorems 2 and 3. If the six-point property fails for some p, the hexagon formulas (26)-(27) collapse. The authors should either prove Lemma 1 or state precisely which theorem of [13] or another reference contains this result.
[§6.4, Theorems 2 and 3] These theorems are not derived. The minimality claims require an argument that the minimum over the moduli space (16) is attained at the endpoint parameters σ = 1 or σ = σ_p. The paper simply concatenates Proposition 5, Corollary 2, and Proposition 6 with formula (22) from [13]. If (22) is accepted as a black box, Theorems 2 and 3 are immediate corollaries, not new results, and the paper should present them as such with the two-line computation. As written, the reader cannot verify the 3Δ and 4Δ factors without the missing proof of Lemma 1 and without an explicit derivation of the endpoint evaluation.
[§5.1, Propositions 3 and 4] The factor 4^m in (17)-(19) is asserted "from Proposition 1 and from [13]", but Proposition 1 only states convexity of 2D. The scaling property d(2^m Λ) = 4^m d(Λ) is standard, but it should be stated explicitly as a lemma with a proof or a precise citation, especially because the paper's later Theorem 1 depends on this scaling law. As written, the transition from (15) to (18) is not justified within the manuscript.

minor comments (5)

[§1, last paragraph] The sentence "In Section we give applications to algebraic number theory" has a missing section number, and no such section appears in the paper.
[§2, equation (14)] The word "hear" should be "here"; the displayed inequality is also not connected to the optimization problem that follows until several paragraphs later.
[§2, Corollary 1] The typo "symmetrc" should be "symmetric"; more importantly, the corollary only concerns convexity and does not by itself justify the 4^m determinant scaling used later.
[§6.4, Theorem 3, formula (27)] The second branch contains an extra comma: "σ p = (2 p − 1)1/p, , 2 ≤ p ≤ p0". The intended condition on p across the two branches should be written more clearly.
[References [14], [15]] The references contain broken LaTeX/encoding, e.g., "Acad/acute.ts1emie"; they should be typeset correctly as "Académie" with proper accents.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the new formulas are scaling and hexagon-area corollaries of the cited critical-determinant theorem, not definitional equivalences or fitted predictions.

full rationale

The derivation chain is not circular. The paper's main new formulas (23)-(24) are immediate consequences of the homogeneity of the critical determinant under dilation: for a planar body, Δ(λD)=λ^2Δ(D), so Δ(2^mD_p)=4^mΔ(D_p); substituting the piecewise expression (22), quoted from the 1986 theorem [13], gives exactly (23)-(24). Theorems 2 and 3 use the elementary identity that the hexagon generated by a lattice basis a1,a2,a1+a2 with the three pairs on the boundary has area 3 det{a1,a2}, so the minimal al-hexagon area is 3Δ(D_p), and the circumscribed case is 4Δ(D_p). These are mathematical deductions from the cited theorem, not definitional tautologies or fitted parameters. The citations [13] and [14] are prior published work (in [13] with co-authors) that states assumptions about the critical determinant of D_p, not about 2^mD_p or hexagon areas, so they are independent evidence under the stated criteria. The paper does contain an unproved assertion (Lemma 1) that the critical lattice shell contains six points, and the sentence in Theorem 1 calling the determinants 'sublattices of index 2^m' is inconsistent with the displayed 4^m factors; but these are correctness and rigor defects, not circularity. No step in the paper reduces its conclusions to its own definitions by construction.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The manuscript introduces the label 'extremal functions on moduli spaces' but this is a re-description of the existing Minkowski-Cohn parameterization of admissible lattices. No new symbolic entity, free parameter, or fitted constant is introduced: p0 and τ_p are defined by equations rather than by data-fitting.

assumptions (4)

domain assumption For each p>1, the critical determinant Δ(D_p) is attained and the critical lattice contains exactly three pairs of boundary points a1, a2, a1+a2.
Invoked in Section 5 and Lemma 1; the moduli-space parameterizations and hexagon-area formulas all depend on the six-point structure of critical lattices, which the paper asserts without proof.
ad hoc to paper The piecewise expression for Δ(D_p) with crossover p0≈2.5725, as quoted in formula (22), is correct.
This is the central computational input, imported from [13]. The paper provides no independent verification, and the threshold p0 is a numerical root defined by equating the two branches.
standard math The scaling law Δ(λK)=λ^n Δ(K) for convex bodies in R^n is valid and applies to D_p.
Used implicitly in Theorem 1 to derive Δ(2^m D_p) from Δ(D_p). The paper never states or proves this.
domain assumption Strict convexity of D_p for p>1 ensures the boundary contains no line segments, so the six-point aligned lattices are the only candidates for the critical lattice.
Mentioned in Section 2 for strictly convex bodies, but the connection to uniqueness of critical lattices is not proved.

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Pith. "Pith review of Extremal functions on moduli spaces and applications." pith.science (2026). https://pith.science/paper/ZW2YNFB6

@misc{pith2026241113671,
  author       = {Pith},
  title        = {Pith review of: Extremal functions on moduli spaces and applications},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ZW2YNFB6}},
  note         = {Machine review of arXiv:2411.13671}
}

abstract

Our object of study is extremal functions which are defined by distance functions of convex bodies. These functions take values in the moduli spaces of algebraic and geometric objects associated with these ${\mathbb Z}$-modules (geometric lattices) and with convex bodies. In most cases, convex bodies are $2$-dimensional Minkowski balls whose boundaries are Minkowski curves and we study lattice points on these curves. We define and investigate extremal functions that yield the homogeneous arithmetic minimum of a function in a lattice, the Hermite constant, the critical determinant of a body, optimal packings of bodies, best values of covering constants, and optimal solutions of Diophantine approximation problems. Moreover, for two-dimensional unit Minkowski balls and Minkowski domains we determine the minimal areas of inscribed and circumscribed hexagons.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

15 extracted references · 15 canonical work pages

[13]

Golovanov, A

Glazunov N., A. Golovanov, A. Malyshev (1986) Proof of Minkows ki’s hy- pothesis about the critical determinant of |x|p + |y|p < 1 domain, Research in Number Theory 9 . Notes of scientiﬁc seminars of LOMI. 151 Leningrad: Nauka. 40–53

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2024

Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.

[1] [13]

Golovanov, A

Glazunov N., A. Golovanov, A. Malyshev (1986) Proof of Minkows ki’s hy- pothesis about the critical determinant of |x|p + |y|p < 1 domain, Research in Number Theory 9 . Notes of scientiﬁc seminars of LOMI. 151 Leningrad: Nauka. 40–53

work page 1986

[2] [14]

On packing of Minkowski balls, Comptes rendus de l’Ac ad/acute.ts1emie bulgare Sci., Tome 76, No 3, 335-342 (2023)

Glazunov N. On packing of Minkowski balls, Comptes rendus de l’Ac ad/acute.ts1emie bulgare Sci., Tome 76, No 3, 335-342 (2023)

work page 2023

[3] [1]

(1907) Diophantische Approximationen , Leipzig: Teubner

Minkowski H. (1907) Diophantische Approximationen , Leipzig: Teubner

work page 1907

[4] [2]

(1941) Lattice points in the region |Ax4| + |By 4| ≥ 1, J

Mordell L.J. (1941) Lattice points in the region |Ax4| + |By 4| ≥ 1, J. London Math. Soc., 16 , 152–156

work page 1941

[5] [3]

(1948) Note on a conjecture by Minkowski, J

Davis C. (1948) Note on a conjecture by Minkowski, J. London Math. Soc. , 23, 172–175

work page 1948

[6] [4]

(1950) Minkowski’s conjectures on critical lattices in the metric {|ξ|p+ |η|p}1/p , Annals of Mathematics, 51 (2), 734–738

Cohn H. (1950) Minkowski’s conjectures on critical lattices in the metric {|ξ|p+ |η|p}1/p , Annals of Mathematics, 51 (2), 734–738

work page 1950

[7] [5]

(1953) Minkowski’s conjecture on the critical lattices of the region |x|p + |y|p ≤ 1 , (I), (II), J

Watson G. (1953) Minkowski’s conjecture on the critical lattices of the region |x|p + |y|p ≤ 1 , (I), (II), J. London Math. Soc. , 28, (3, 4), 305–309, 402–410

work page 1953

[8] [6]

Cassels J. W. S. (1997) An Introduction to the Geometry of Num bers, NY, Springer

work page 1997