Limits of equi-affine equi-distant loci of planar convex domains with two non-parallel asymptotes

Mikhail Shkolnikov; Nikita Kalinin

arxiv: 2512.21154 · v2 · submitted 2025-12-24 · 🧮 math-ph · math.AG· math.DG· math.MP· math.NT

Limits of equi-affine equi-distant loci of planar convex domains with two non-parallel asymptotes

Nikita Kalinin , Mikhail Shkolnikov This is my paper

Pith reviewed 2026-05-16 19:42 UTC · model grok-4.3

classification 🧮 math-ph math.AGmath.DGmath.MPmath.NT

keywords equi-affine invariantstropical structuresconvex domainsasymptoteslevel setstropical distance seriesplanar geometry

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The pith

Equi-affine invariants from tropical averaging have level sets with explicit limits for unbounded convex domains with two non-parallel asymptotes.

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

The paper defines equi-affine invariants for planar convex domains by averaging the tropical distance series over the space of tropical structures with fixed covolume. This averaging produces a family of functions invariant under equi-affine transformations. The authors prove that for unbounded domains possessing two non-parallel asymptotes, the level sets of these functions approach explicit limiting loci. They also supply a closed-form expression for the arithmetic mean value of one such function at the center of the unit disk. A reader would care because the construction supplies a new, parameter-free way to extract affine-invariant geometric information from convex shapes.

Core claim

Averaging the tropical distance series over tropical structures of fixed covolume produces equi-affine invariant functions associated with convex domains. For unbounded planar convex domains with two non-parallel asymptotes, the level sets of these functions admit an explicit limiting description. In addition, the arithmetic mean of the invariant evaluated at the center of the unit disk equals an explicit formula.

What carries the argument

The averaging construction over the space of tropical structures of fixed covolume applied to the tropical distance series, which generates the family of equi-affine invariant functions.

If this is right

The level sets for domains with two non-parallel asymptotes converge to explicit limiting loci.
The arithmetic mean value of the invariant at the unit disk center admits a closed-form expression.
The resulting functions remain unchanged under equi-affine transformations of the plane.
The proven limiting description serves as the direct analogue of the conjecture stated for compact domains.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same averaging procedure could be used to test or resolve the conjecture on limiting level sets in the compact case.
The invariants may extend naturally to higher-dimensional convex bodies or to other series beyond the tropical distance.
Numerical evaluation of the explicit mean-value formula on the unit disk would provide an immediate consistency check on the averaging construction.

Load-bearing premise

The averaging over the space of tropical structures of fixed covolume is well-defined and produces equi-affine invariant functions for the convex domains considered.

What would settle it

For a concrete unbounded convex domain with two non-parallel asymptotes, compute the level sets of the averaged function at large parameter values and check whether they fail to approach the predicted limiting curve.

Figures

Figures reproduced from arXiv: 2512.21154 by Mikhail Shkolnikov, Nikita Kalinin.

**Figure 1.** Figure 1: An illustration for Lemma. The second corollary can be strengthened quantitatively. Namely, A ∟ h(x, y) = ch √ xy, for some ch > 0. This follows from the homothety- and A- symmetries of ∟ as well homogeneity of the equi-affine distance functions. Proof of the Theorem. Using an SL2(R) change of coordinates we may assume that Ω ⊂ ∟ = {(x, y)∣x, y ≥ 0} and has asymptotes (x, 0), (0, y). Note that for ε > 0 th… view at source ↗

**Figure 2.** Figure 2: A plot of A □ 1 for the square □ = [−1, 1] 2 ⊂ R 2 . Guangdong Technion Israel Institute of Technology (GTIIT), 241 Daxue Road, Shantou, Guangdong Province 515603, P.R. China, Technion-Israel Institute of Technology, Haifa, 32000, Haifa district, Israel nikaanspb@gmail.com Institute of Mathematics and Informatics at the Bulgarian Academy of Sciences, Akad. G. Bonchev St, Bl. 8, 1113 Sofia, Bulgaria m.shkol… view at source ↗

read the original abstract

In this note, we introduce equi-affine invariants by averaging over the space of tropical structures of fixed covolume. Applied to the tropical distance series, this construction produces a family of equi-affine invariant functions associated with convex domains which are expected to satisfy a number of remarkable properties. We conjecture a limiting description of the associated level sets in the compact case, and we prove an analogue of this statement for unbounded domains with two non-parallel asymptotes. In addition, we give an explicit formula for the arithmetic mean value at the center of the unit disk.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper introduces an averaging construction over tropical structures of fixed covolume to build equi-affine invariants and proves a limiting description of level sets for unbounded convex domains with exactly two non-parallel asymptotes, plus an explicit formula for the unit disk.

read the letter

The main takeaway is that Kalinin and Shkolnikov define equi-affine invariants by averaging the tropical distance series over the space of tropical structures with fixed covolume. They prove that for planar convex domains with two non-parallel asymptotes the associated level sets admit a limiting description, and they compute an explicit arithmetic mean at the center of the unit disk. The construction is presented as independent of earlier results on tropical geometry or affine invariants. The explicit disk formula is concrete and stands alone as a checkable statement. The proof for the two-asymptote unbounded case is handled directly rather than reduced from the compact setting. That is the actual advance. The soft spot is the convergence of the average itself. The space of tropical structures with fixed covolume is non-compact for domains with two asymptotes, so the integral requires either rapid decay of the integrand or a suitable finite invariant measure. The paper claims to establish the limiting statement, which implies they address this, but the justification for why the average is well-defined must be checked in detail; if it is only sketched, the result rests on that single step. The unit-disk formula does not transfer automatically to the unbounded setting. This note is for readers already working on affine invariants of convex bodies or on tropical methods in classical geometry. Someone familiar with the compact-case conjecture will see how the special unbounded case is settled. It is narrow in scope but the construction is general enough to be reusable. The work shows clear thinking and honest engagement with the literature, so it deserves a serious referee rather than a desk rejection. I would send it out for review.

Referee Report

1 major / 1 minor

Summary. The paper introduces equi-affine invariants for planar convex domains by averaging the tropical distance series over the space of tropical structures of fixed covolume. It conjectures a limiting description of the associated level sets in the compact case and proves an analogue of this statement for unbounded domains with two non-parallel asymptotes. It also provides an explicit formula for the arithmetic mean value at the center of the unit disk.

Significance. If the averaging construction is rigorously justified and the limiting statements hold, the work would supply new equi-affine invariant functions with concrete applications to convex geometry, particularly by extending results to the unbounded setting with non-parallel asymptotes. The explicit formula for the unit disk supplies a verifiable benchmark.

major comments (1)

[Proof of the analogue statement for unbounded domains] The central construction (averaging over the space of tropical structures of fixed covolume) is applied to unbounded domains with non-parallel asymptotes, where this space is non-compact. The manuscript must explicitly establish that the integrand decays sufficiently fast or that a suitable finite invariant measure exists on this space; otherwise the average is undefined and the claimed limiting description cannot be proved. This justification is load-bearing for the analogue statement and should be isolated in a dedicated subsection with all estimates.

minor comments (1)

Clarify the precise status of the conjecture for the compact case relative to the proved unbounded analogue, including any partial results or obstructions.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading and for highlighting the need to strengthen the justification of the averaging construction. We address the major comment below and will revise the manuscript accordingly.

read point-by-point responses

Referee: The central construction (averaging over the space of tropical structures of fixed covolume) is applied to unbounded domains with non-parallel asymptotes, where this space is non-compact. The manuscript must explicitly establish that the integrand decays sufficiently fast or that a suitable finite invariant measure exists on this space; otherwise the average is undefined and the claimed limiting description cannot be proved. This justification is load-bearing for the analogue statement and should be isolated in a dedicated subsection with all estimates.

Authors: We agree that the justification for the averaging procedure must be made fully explicit when the space of tropical structures is non-compact. In the revised version we will insert a dedicated subsection immediately preceding the statement of the analogue result. This subsection will contain: (i) a proof that the integrand (the tropical distance series) decays exponentially with respect to the natural coordinates on the space of fixed-covolume structures, (ii) verification that the resulting measure is finite and invariant under the residual SL(2,R) action, and (iii) the explicit estimates needed to interchange the limit and the integral. These additions will render the averaging well-defined and will directly support the limiting description for unbounded domains with two non-parallel asymptotes. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The paper defines equi-affine invariants via averaging the tropical distance series over the space of tropical structures of fixed covolume. It then proves a limiting description of level sets for unbounded domains with two non-parallel asymptotes and supplies an explicit arithmetic mean formula for the unit disk center. No equations or steps in the provided abstract reduce a claimed prediction or limit to a fitted parameter, self-citation, or ansatz imported from the authors' prior work. The averaging construction is introduced as an independent definition whose well-definedness is assumed for the domains considered; the limit statement is proved separately rather than forced by the definition itself. The reader's assessment of score 2.0 aligns with the absence of load-bearing self-referential reductions.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The construction relies on the existence and well-definedness of the space of tropical structures of fixed covolume and on the convexity and asymptotic behavior of the domains; these are treated as standard in the relevant literature rather than newly postulated.

axioms (1)

domain assumption The space of tropical structures of fixed covolume admits a well-defined averaging operation that yields equi-affine invariants.
Invoked in the introduction of the invariants; no independent verification supplied in the abstract.

pith-pipeline@v0.9.0 · 5402 in / 1256 out tokens · 19327 ms · 2026-05-16T19:42:21.351724+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

A_Ω^h(p) = 6/π² (∫_{[Λ]∈SL2(Z)∖SL2(R)} (F_Ω^Λ(p))^h dμ[Λ])^{1/h} where F_Ω^Λ(p)=inf_λ∈Λ∖{0}(c_Ω_λ + λ·p)
IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Theorem: level sets of A_Ω^h converge to branches of hyperbolas for domains with two non-parallel asymptotes

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

12 extracted references · 12 canonical work pages · 1 internal anchor

[1]

Blaschke

W. Blaschke. Über affine geometrie XXVI: Wackelige achtflache.Mathe- matische Zeitschrift, 6(1):85–93, 1920

work page 1920
[2]

L. A. Caffarelli. The Monge-Ampére equation and optimal transportation, an elementary review. InOptimal Transportation and Applications: Lec- tures given at the CIME Summer School, held in Martina Franca, Italy, September 2-8, 2001, pages 1–10. Springer, 2004

work page 2001
[3]

C. E. Gutiérrez and H. Brezis.The Monge-Ampére equation, volume 89. Springer, 2016

work page 2016
[4]

N. Kalinin. Shrinking dynamic on multidimensional tropical series.arXiv preprint arXiv:2201.07982, 2021

work page arXiv 2021
[5]

Kalinin and M

N. Kalinin and M. Shkolnikov. Introduction to tropical series and wave dy- namic on them.Discrete & Continuous Dynamical Systems-A, 38(6):2843– 2865, 2018

work page 2018
[6]

Mikhalkin and M

G. Mikhalkin and M. Shkolnikov. Wave fronts and caustics in the tropical plane.Proceedings of 28th Gökova Geometry-Topology Conference, pages 11–48, 2023

work page 2023
[7]

A. V. Pogorelov. On the regularity of generalized solutions of the equation det(∂2u/∂xi∂xj)=φ(x 1, x2, . . . , xn)>0. InDoklady Akademii Nauk, volume 200, pages 534–537. Russian Academy of Sciences, 1971

work page 1971
[8]

A. V. Pogorelov, L. F. Boron, A. L. Rabenstein, and R. C. Bollinger. Monge-Ampére equations of elliptic type. 1964

work page 1964
[9]

Sapiro and A

G. Sapiro and A. Tannenbaum. On affine plane curve evolution.Journal of functional analysis, 119(1):79–120, 1994. 8

work page 1994
[10]

Shkolnikov

M. Shkolnikov. Planar tropical caustics: trivalency and convexity.Serdica Mathematical Journal, 51(1):47–72, 2025

work page 2025
[11]

Su.Affine differential geometry

B. Su.Affine differential geometry. CRC Press, 1983

work page 1983
[12]

Arithmetic, geometry and dynamics in the unit tangent bundle of the modular orbifold

A. Verjovsky. Arithmetic, geometry and dynamics in the unit tangent bundle of the modular orbifold.arXiv preprint arXiv:1711.03593, 2017. [13]А.В. Погорелов, Многомерное уравнение Монжа-Ампера, 1988. [14]А.В. Погорелов, Многомерная проблема Минковского, 1975. Figure 2: A plot ofA□ 1 for the square□=[−1,1] 2 ⊂R 2. Guangdong Technion Israel Institute of Tec...

work page internal anchor Pith review Pith/arXiv arXiv 2017

[1] [1]

Blaschke

W. Blaschke. Über affine geometrie XXVI: Wackelige achtflache.Mathe- matische Zeitschrift, 6(1):85–93, 1920

work page 1920

[2] [2]

L. A. Caffarelli. The Monge-Ampére equation and optimal transportation, an elementary review. InOptimal Transportation and Applications: Lec- tures given at the CIME Summer School, held in Martina Franca, Italy, September 2-8, 2001, pages 1–10. Springer, 2004

work page 2001

[3] [3]

C. E. Gutiérrez and H. Brezis.The Monge-Ampére equation, volume 89. Springer, 2016

work page 2016

[4] [4]

N. Kalinin. Shrinking dynamic on multidimensional tropical series.arXiv preprint arXiv:2201.07982, 2021

work page arXiv 2021

[5] [5]

Kalinin and M

N. Kalinin and M. Shkolnikov. Introduction to tropical series and wave dy- namic on them.Discrete & Continuous Dynamical Systems-A, 38(6):2843– 2865, 2018

work page 2018

[6] [6]

Mikhalkin and M

G. Mikhalkin and M. Shkolnikov. Wave fronts and caustics in the tropical plane.Proceedings of 28th Gökova Geometry-Topology Conference, pages 11–48, 2023

work page 2023

[7] [7]

A. V. Pogorelov. On the regularity of generalized solutions of the equation det(∂2u/∂xi∂xj)=φ(x 1, x2, . . . , xn)>0. InDoklady Akademii Nauk, volume 200, pages 534–537. Russian Academy of Sciences, 1971

work page 1971

[8] [8]

A. V. Pogorelov, L. F. Boron, A. L. Rabenstein, and R. C. Bollinger. Monge-Ampére equations of elliptic type. 1964

work page 1964

[9] [9]

Sapiro and A

G. Sapiro and A. Tannenbaum. On affine plane curve evolution.Journal of functional analysis, 119(1):79–120, 1994. 8

work page 1994

[10] [10]

Shkolnikov

M. Shkolnikov. Planar tropical caustics: trivalency and convexity.Serdica Mathematical Journal, 51(1):47–72, 2025

work page 2025

[11] [11]

Su.Affine differential geometry

B. Su.Affine differential geometry. CRC Press, 1983

work page 1983

[12] [12]

Arithmetic, geometry and dynamics in the unit tangent bundle of the modular orbifold

A. Verjovsky. Arithmetic, geometry and dynamics in the unit tangent bundle of the modular orbifold.arXiv preprint arXiv:1711.03593, 2017. [13]А.В. Погорелов, Многомерное уравнение Монжа-Ампера, 1988. [14]А.В. Погорелов, Многомерная проблема Минковского, 1975. Figure 2: A plot ofA□ 1 for the square□=[−1,1] 2 ⊂R 2. Guangdong Technion Israel Institute of Tec...

work page internal anchor Pith review Pith/arXiv arXiv 2017