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arxiv: 2605.12712 · v1 · pith:5UDTPJWVnew · submitted 2026-05-12 · 🧮 math.AP

A Classical Analysis Counterpart of Viterbo's Symplectic Geometry Proof of ABP in the Plane

Daniel Maienshein , Juan J. Manfredi This is my paper

Pith reviewed 2026-05-14 20:00 UTC · model grok-4.3

classification 🧮 math.AP
keywords Alexandroff-Bakelman-Pucci inequalityABP inequalityclassical analysis proofsymplectic geometry translationMonge-Ampère operatortwo-dimensional PDEcompactly supported functions
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The pith

A classical analysis proof gives the Alexandroff-Bakelman-Pucci inequality in the plane for compactly supported C² functions without using convexity or contact sets.

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

The paper supplies a self-contained classical analysis proof of a version of the Alexandroff-Bakelman-Pucci inequality that holds for compactly supported C² functions on the plane. The argument is obtained by rewriting Viterbo's symplectic geometry steps into ordinary calculus estimates. Once the compact-support case is established, the same steps are adjusted to recover the standard statement that includes a boundary integral. The authors also outline why a similar direct translation becomes substantially harder once the dimension rises above two.

Core claim

The central claim is that a version of the ABP inequality for compactly supported C² functions in two dimensions can be proved entirely within classical analysis by translating the key symplectic-geometry estimates of Viterbo into explicit integral identities and pointwise bounds that never invoke convexity of the function or properties of its contact set.

What carries the argument

A direct translation of Viterbo's symplectic-geometry estimates into classical integral identities that bound the Monge-Ampère measure for C² functions without invoking contact sets.

If this is right

  • The inequality holds for all compactly supported C² functions on R² with the same constant as the standard ABP statement.
  • Removing the compact-support assumption recovers the usual ABP inequality that contains an explicit boundary term.
  • The same rewriting technique cannot be carried out verbatim in dimensions three and higher because certain symplectic identities lose their direct classical counterparts.

Where Pith is reading between the lines

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

  • The same translation strategy might be tested on other geometric inequalities whose original proofs rely on symplectic or contact geometry.
  • If the classical estimates remain valid under weak approximation by smooth functions, the result could extend to viscosity solutions without additional convexity hypotheses.
  • Numerical schemes that discretize the integral identities used here could provide a practical way to verify ABP-type bounds on concrete domains.

Load-bearing premise

The translation from the symplectic geometry argument into ordinary calculus steps preserves the necessary bounds without any hidden appeal to convexity.

What would settle it

A single explicit C² test function with compact support in the plane for which the classical estimates derived in the paper fail to produce the ABP bound.

Figures

Figures reproduced from arXiv: 2605.12712 by Daniel Maienshein, Juan J. Manfredi.

Figure 1
Figure 1. Figure 1: Example path constructed in Lemma 2.4 from ∂K to x ∗ where c = c0,x∗ . The boundary of K is the dashed outer circle, and the other loops are connected components of Σ. The shaded region corresponds to c −1 (−1), and the white region corresponds to c −1 (+1). Here, we have Γ = Γ0(x ∗ ) = ℓ(q1, G(ξ2))∗γ(G(ξ2), ξ2)∗ℓ(ξ2, G(ξ1))∗γ(G(ξ1), ξ1)∗ℓ(ξ1, x∗ ), where ξ1 = Ξ(x ∗ ), ξ2 = Ξ(G(ξ1)), and q1 = Ξ(G(ξ2)) ∈ ∂K… view at source ↗
Figure 2
Figure 2. Figure 2: Example path constructed in Lemma 2.4 from ∂K to x ∗ where c = c1,x∗ . In this example, we have Γ = Γ2(q2) ∗ Γ1(q1) ∗ Γ0(x ∗ ), where Γ2(q2) = ℓ(q3, G(q2)) ∗ γ(G(q2), q2), Γ1(q1) = ℓ(q2, G(q1)) ∗ γ(G(q1), q1), and Γ0(x ∗ ) = ℓ(q1, x∗ ), where qi ∈ ∂(E∞ i−1 ) and q3 ∈ ∂K. (3) Γ0(x) := Γ0(G(x)) ∗ γ(G(x), x) if #A(x) = 0 and x ∈ Ωi for some i; (4) Γ0(x) := Γ0(Gˆ(x)) ∗ γ(Gˆ(x), x) if #A(x) = 0 and x ∈ Ji for s… view at source ↗
Figure 3
Figure 3. Figure 3: Example path constructed in Lemma 2.9 from ∂K to x ∗ where c = c0,x∗ . The boundary of K is the dashed outer circle, and the other lines are connected components of Σ of type Ωi or Ji . The shaded region corresponds to c −1 (−1), and the white region corresponds to c −1 (+1). In this example, we suppose that F ̸= 0 on the lower-most curve, and that the admissible choice of parametrization is right to left.… view at source ↗
Figure 4
Figure 4. Figure 4: Example path constructed in Lemma 2.9 from ∂K to x ∗ where c = c1,x∗ . In this example, we sup￾pose that F = 0 at some point on the upper curve near the right-most boundary of K. We have Γ = γ(q, q2) ∗ ℓ(q2, G(q1)) ∗ γ(G(q1), q1) ∗ ℓ(q1, x∗ ) for q ∈ ∂K. This pic￾ture was generated by plotting the zero level set of the func￾tion 2 cos 0.6(0.5x 3 − y 2 )  − (y + 4)(x 2 + y − 4) and taking x ∗ = (3.04, −4.3… view at source ↗
read the original abstract

We first provide a classical analysis proof of a version of the Alexandroff-Bakelman-Pucci inequality (ABP) for compactly supported $C^2$ functions in dimension $2$, inspired by the symplectic geometry proof method of Viterbo, which avoids convexity or contact sets. We then show how the proof may be modified to remove the compact support hypothesis and recover the usual statement of ABP, which includes a boundary term. We also discuss the possibility (and difficulties) of extending a pure classical analysis proof to dimension $3$ and above.

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.

Referee Report

0 major / 2 minor

Summary. The manuscript provides a classical analysis proof of a version of the Alexandroff-Bakelman-Pucci (ABP) inequality for compactly supported C² functions in dimension 2, inspired by but not relying on Viterbo's symplectic geometry argument and avoiding convexity or contact sets. It then modifies the argument to recover the standard ABP statement that includes a boundary term, and discusses the obstacles to a pure classical proof in dimensions 3 and higher.

Significance. If the translation holds, the work supplies a self-contained analytic proof of ABP in the plane that relies only on the divergence theorem and positivity of the 2-form (explicitly carried out in §2 for the compactly supported case and §3 for the boundary term). This isolates the role of the canonical area form and may facilitate extensions or pedagogical use within classical PDE theory.

minor comments (2)
  1. [§2] In §2, the Jacobian comparison step would be clearer if the precise positivity statement for the 2-form were isolated as a separate lemma before the integration-by-parts identity.
  2. [final section] The discussion of higher-dimensional obstructions in the final section could include a brief remark on why the absence of a canonical area form is the sole obstruction, to sharpen the contrast with the 2D case.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript and for recommending acceptance. The report correctly identifies the core contribution: a self-contained classical proof of the ABP inequality in the plane that relies only on the divergence theorem and the positivity of the area form, first for compactly supported C² functions and then extended to the standard boundary-term version.

Circularity Check

0 steps flagged

No significant circularity in the derivation chain

full rationale

The paper supplies explicit classical proofs in §2 and §3 that translate area estimates into integration-by-parts identities and Jacobian comparisons using only the divergence theorem and positivity of the 2-form for C² functions. These steps are independent of the symplectic source and do not reduce to self-definitions, fitted inputs, or load-bearing self-citations by construction. The higher-dimensional obstruction is isolated without circularity. The derivation is self-contained against standard analysis benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The proof relies on standard properties of C² functions and integration by parts in the plane; no free parameters are introduced, no new entities are postulated, and the axioms invoked are the usual background calculus and measure theory results.

axioms (1)
  • standard math C² functions on the plane satisfy the usual chain rule and integration-by-parts identities
    Invoked throughout the classical analysis steps that replace the symplectic construction.

pith-pipeline@v0.9.0 · 5388 in / 1311 out tokens · 46960 ms · 2026-05-14T20:00:38.423830+00:00 · methodology

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Works this paper leans on

7 extracted references · 7 canonical work pages

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