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arxiv: 2401.10992 · v2 · submitted 2024-01-19 · 🧮 math.FA · math.CV

Two-dimensional Bl{}ocki, L^p-Mahler, and Bourgain conjectures

Vlassis Mastrantonis , Yanir A. Rubinstein This is my paper

Pith reviewed 2026-05-24 04:49 UTC · model grok-4.3

classification 🧮 math.FA math.CV
keywords L^p-Mahler conjectureBlocki conjectureBourgain hyperplane conjecturetube domainsconvex polytopesisotropic constantvertex slidingpolar body
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The pith

In two dimensions the L^p-Mahler conjectures hold for all p, confirming Blocki's conjectures on Bergman kernels and Bourgain's hyperplane conjecture.

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

The paper proves that the L^p-Mahler volume conjectures hold for every real number p when the ambient space is two-dimensional. The case p=1 recovers Blocki's conjectures that give sharp lower bounds for Bergman kernels on tube domains. The case p=∞ recovers the classical Mahler conjectures. The argument follows the change in the L^p-polar body when the vertices of a two-dimensional polytope are slid in the classical Mahler manner. The same calculation shows that the isotropic constant raised to a suitable power is a convex quadratic polynomial in the sliding parameter, which immediately yields an elementary proof of Bourgain's strong hyperplane conjecture in dimension two.

Core claim

We confirm the two-dimensional L^p-Mahler conjectures by studying the effect of Mahler's sliding of vertices on the L^p-polar body. This establishes the conjectures for p=1 which are Blocki's conjectures on Bergman kernels of tube domains and for p=∞ which are Mahler's conjectures. The same technique yields an elementary proof of Bourgain's strong hyperplane conjectures in dimension two by showing that the isotropic constant to a suitable power is a convex quadratic polynomial in the sliding parameter.

What carries the argument

Mahler's vertex sliding applied to the L^p-polar body of two-dimensional polytopes

If this is right

  • The Bergman kernels of two-dimensional tube domains satisfy the sharp lower bounds conjectured by Blocki.
  • The classical Mahler volume conjecture holds in dimension two as the p=∞ case of the L^p-Mahler conjectures.
  • Bourgain's strong hyperplane conjecture holds in dimension two.
  • The isotropic constant raised to an appropriate power is a convex quadratic polynomial in the sliding parameter.

Where Pith is reading between the lines

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

  • If analytic control of the L^p-polar body under sliding can be obtained in higher dimensions, the same conjectures would hold there as well.
  • The sliding technique may apply to other functionals on convex bodies that arise in functional analysis or asymptotic convex geometry.
  • The explicit quadratic form found for the isotropic constant under sliding suggests that similar low-degree polynomials may govern other extremal quantities in two-dimensional convex geometry.

Load-bearing premise

The effect of sliding vertices on two-dimensional polytopes can be tracked analytically on the L^p-polar body even though duality is lost and the non-symmetric case is more involved.

What would settle it

A single two-dimensional convex body for which the L^p-Mahler volume inequality fails for some real p, or for which the isotropic constant raised to the appropriate power fails to be convex under vertex sliding.

Figures

Figures reproduced from arXiv: 2401.10992 by Vlassis Mastrantonis, Yanir A. Rubinstein.

Figure 1
Figure 1. Figure 1: |K◦,p| = |K◦,p ∩ {x ≥ 0}| + |K◦,p ∩ {x ≤ 0}|. Observe these two areas are equal in the symmetric case. Proof. Starting with (1.7), in polar coordinates, |K◦,p| = 1 2 Z R2 e −hp,K(x,y) dxdy = 1 2 Z ∂B2 2 Z ∞ 0 e −hp,K(ru) rdr du = 1 2 Z ∂B2 2 du kuk 2 K◦,p , 10 [PITH_FULL_IMAGE:figures/full_fig_p010_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Symmetric Mahler sliding eliminating two antipodal vertices simultaneously. • reduces the L p -Mahler volume, Before proceeding, let us fix some notation. Let S := co {(x1, y1), . . . ,(xm, ym), −(x1, y1), . . . , −(xm, ym)} denote a symmetric polytope with 2m vertices. After rotating S, and possibly renaming the vertices, we can assume that (x1, y1), . . . ,(xm, ym) are counter-clockwise oriented, y2 > 0,… view at source ↗
Figure 3
Figure 3. Figure 3: Mahler’s sliding eliminating a vertex The following generalizes Proposition 3.1. In comparison to the symmetric case, the non￾symmetric case requires a combination of translation and sliding (Figures 3–4). 13 [PITH_FULL_IMAGE:figures/full_fig_p013_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Sliding with translation Proposition 5.6. Let P ∈ P with 0 ∈ int P. For any x2, x′ 2 ∈ [ξℓ , ξr] (recall [PITH_FULL_IMAGE:figures/full_fig_p024_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: When P mostly lies in {x ≥ 0} then P ◦ mostly lies in {x ≤ 0}. Several basic facts in this direction are the following: 24 [PITH_FULL_IMAGE:figures/full_fig_p024_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: α and β Lemma 5.10. Let p ∈ (0,∞]. Let K ⊂ R 2 be a convex body with 0 ∈ int K. For α := min{x ∈ R : (x, y) ∈ K} and β := max{x ∈ R : (x, y) ∈ K}, |(K − (α, 0))◦,p ∩ {x > 0}|, |(K − (β, 0))◦,p ∩ {x < 0}| < ∞, |(K − (α, 0))◦,p ∩ {x < 0}| = |(K − (β, 0))◦,p ∩ {x > 0}| = ∞, Proof. Since K − (α, 0) ⊂ {x ≥ 0}, Claim 5.7 implies |(K − (α, 0))◦,p| = ∞, while Claim 5.8 implies |(K − (α, 0))◦,p ∩ {x ≥ 0}| < ∞, so n… view at source ↗
Figure 7
Figure 7. Figure 7: α ′ < α as it relies on a Santal´o-type argument not specific to planar bodies. It also could be used to prove [41, Lemma 5]. 5.4 Proof of Theorem 1.10 Proof of Lemma 5.1. Let x2, x′ 2 ∈ [ξℓ , ξr] and recall the notation (5.7). By Proposition 5.6, there exists x0 ∈ R with (x0, 0),(−x0, 0) ∈ int Pb, and I+(x2, x0) I−(x2, x0) = I+(x ′ 2 , −x0) I−(x ′ 2 , −x0) =: ρ. (5.20) Thus, |P((x2) − (x0, 0))◦,p| = (1 + … view at source ↗
Figure 8
Figure 8. Figure 8: ∆l,h(x2). 34 [PITH_FULL_IMAGE:figures/full_fig_p034_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Pb remains isotropic after roation. By Corollary 6.14, for P ∈ Pe, |P| 4 C(P(x2)) = C(Pb) −1 + C(Pb) − 1 2  |∆|tr Cov(∆(x2)) + |∆| |P| tr b(∆(x2))b(∆(x2))T  + |∆| 4 C(∆(x2)) + |∆| 2 |P| b N (∆(x2))T Cov(∆(x2))b N (∆(x2)). (6.11) Thus, to prove Lemma 6.3 it is enough to compute the components of (6.11). In the notation of §6.4, by Lemma 6.15, tr Cov(∆l,h(x2) + (x0, y0)) = tr Cov(∆l,h(x2)) = x 2 2 18 + h 2… view at source ↗
read the original abstract

We confirm, in dimension two, Blocki's conjectures on sharp lower bounds for Bergman kernels of tube domains. To that end, we verify a broader class of $L^p$-Mahler conjectures due to Berndtsson and the authors, where $p=1$ are Blocki's conjecture, and $p=\infty$ are Mahler's conjectures. The proofs are technically challenging as the $L^p$-Mahler volume is considerably harder to deal with analytically compared to Mahler's volume, and furthermore duality is lost. In addition, unlike in the classical Mahler setting, the non-symmetric setting is considerably more involved than the symmetric one. The proofs involve studying the effect of Mahler's classical sliding of vertices on two-dimensional polytopes on the $L^p$-polar body (no longer a polytope). Some arguments are inspired by works of Campi--Gronchi and Meyer--Reisner on volumes of classical polar bodies of shadow systems. In passing, we also explore how Mahler's sliding affects the isotropic constant. This leads to an elementary proof of Bourgain's strong hyperplane conjectures in dimension two, originally due to Bisztriczky--B\"or\"oczky, Campi--Colesanti--Gronchi and Meckes. Specifically, we show that, as a function of the sliding parameter, the isotropic constant raised to an appropriate power is a convex quadratic polynomial.

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

2 major / 2 minor

Summary. The manuscript confirms the two-dimensional cases of Błocki's conjectures on sharp lower bounds for Bergman kernels of tube domains by verifying the L^p-Mahler conjectures (p=1 recovers Błocki; p=∞ recovers Mahler). Proofs proceed by analyzing the effect of Mahler's classical vertex sliding on 2D polytopes upon the L^p-polar body (no longer a polytope). In passing, the isotropic constant raised to a suitable power is shown to be a convex quadratic polynomial in the sliding parameter, yielding an elementary proof of Bourgain's strong hyperplane conjecture in dimension two.

Significance. If the central claims hold, the work resolves several long-standing conjectures in convex geometry and complex analysis in the planar case. The elementary quadratic-polynomial argument for the isotropic constant and the extension of sliding techniques to L^p-Mahler volumes (despite loss of duality and greater difficulty in the non-symmetric setting) are technically noteworthy and may inform higher-dimensional approaches.

major comments (2)
  1. [sliding argument for non-symmetric polytopes] The load-bearing step is the analytic control of the L^p-Mahler volume functional under vertex sliding for non-symmetric 2D polytopes (abstract and the section developing the sliding argument). Because the L^p-polar body is not a polytope, duality is unavailable and separate estimates are required; the manuscript must supply the explicit first derivative (or monotonicity/convexity statement) with respect to the sliding parameter together with boundary analysis to guarantee that the minimum occurs at the conjectured extremizer.
  2. [isotropic-constant paragraph] The claim that the isotropic constant to an appropriate power is a convex quadratic polynomial in the sliding parameter (the paragraph beginning 'In passing, we also explore...') is used to obtain the elementary proof of the 2D hyperplane conjecture. The coefficients of this quadratic must be computed explicitly and shown to be non-negative to confirm convexity; without this, the reduction to the known extremal cases is incomplete.
minor comments (2)
  1. Clarify the precise range of p for which the L^p-Mahler conjectures are proved and state whether the arguments are uniform in p or require separate treatment at the endpoints p=1 and p=∞.
  2. Expand the discussion of the inspiration from Campi–Gronchi and Meyer–Reisner to indicate which specific shadow-system estimates are adapted and where the new difficulties arising from the L^p setting appear.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and valuable comments on our manuscript. We address each major comment below and will revise the paper accordingly to provide the requested explicit details.

read point-by-point responses

  1. Referee: [sliding argument for non-symmetric polytopes] The load-bearing step is the analytic control of the L^p-Mahler volume functional under vertex sliding for non-symmetric 2D polytopes (abstract and the section developing the sliding argument). Because the L^p-polar body is not a polytope, duality is unavailable and separate estimates are required; the manuscript must supply the explicit first derivative (or monotonicity/convexity statement) with respect to the sliding parameter together with boundary analysis to guarantee that the minimum occurs at the conjectured extremizer.

    Authors: We agree that an explicit derivative computation would strengthen the presentation. The section developing the sliding argument already derives the monotonicity of the L^p-Mahler volume functional via direct differentiation with respect to the sliding parameter and analyzes the boundary behavior to locate the minimum. In the revision we will extract and display the explicit first-derivative formula together with the boundary analysis in a dedicated lemma, making the control fully transparent. revision: yes

  2. Referee: [isotropic-constant paragraph] The claim that the isotropic constant to an appropriate power is a convex quadratic polynomial in the sliding parameter (the paragraph beginning 'In passing, we also explore...') is used to obtain the elementary proof of the 2D hyperplane conjecture. The coefficients of this quadratic must be computed explicitly and shown to be non-negative to confirm convexity; without this, the reduction to the known extremal cases is incomplete.

    Authors: We accept the referee's observation. While the manuscript establishes that the relevant power of the isotropic constant is a quadratic polynomial in the sliding parameter, the coefficients are not written out explicitly. In the revised version we will compute these coefficients in closed form, verify that each is non-negative, and thereby complete the convexity argument and the reduction to the extremal cases. revision: yes

Circularity Check

0 steps flagged

Direct analytic verification of L^p-Mahler conjectures via vertex sliding; self-contained with external inspirations

full rationale

The paper derives its confirmations of the Blocki and L^p-Mahler conjectures (and the Bourgain result) by explicitly analyzing the variation of the L^p-polar volume under Mahler's vertex sliding on 2D polytopes, showing the minimum occurs at the conjectured position. This rests on direct computations of the functional's derivative and convexity properties with respect to the sliding parameter, plus an explicit verification that the isotropic constant (to a power) is a convex quadratic polynomial in that parameter. Arguments draw inspiration from the external Campi-Gronchi and Meyer-Reisner works on classical polars; the sole self-reference attributes the conjecture statement itself rather than supplying any load-bearing step of the proof. No parameters are fitted to data, no quantity is renamed as a prediction, and no uniqueness or ansatz is imported via self-citation chain. The derivation is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract only; no information on free parameters, axioms or invented entities is supplied.

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