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arxiv: 2605.09334 · v2 · pith:5H3E7PVCnew · submitted 2026-05-10 · 🧮 math.MG

The non-symmetric Mahler conjecture in dimension three

Shibing Chen , Yuanyuan Li , Dongmeng Xi , Zhefeng Xu This is my paper

Pith reviewed 2026-05-22 10:46 UTC · model grok-4.3

classification 🧮 math.MG
keywords Mahler conjecturenon-symmetric volume productSantaló pointconvex bodiesthree dimensionsvolume inequalitypolar body
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The pith

Every convex body in three dimensions has non-symmetric volume product at least 64/9 at its Santaló point.

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

The paper establishes a sharp lower bound for the non-symmetric volume product of convex bodies in three-dimensional space. It shows that this product, defined with respect to the Santaló point, is never smaller than 64/9. A sympathetic reader would care because this resolves the non-symmetric Mahler conjecture in dimension three, providing a precise minimal value for this measure of volume asymmetry in convex geometry. The result builds on prior work in two dimensions and offers a concrete quantitative statement about how volumes of a body and its polar relate.

Core claim

We prove the sharp lower bound P(K) ≥ 64/9 for every convex body K in R^3, where P(K) denotes the non-symmetric volume product with respect to the Santaló point.

What carries the argument

The non-symmetric volume product P(K) defined using the Santaló point as the reference for the polar body.

If this is right

  • The bound is achieved by certain convex bodies, confirming sharpness.
  • The Mahler volume inequality holds in its non-symmetric form for all 3D convex sets.
  • This provides the exact minimal constant for the product of volumes in 3D.

Where Pith is reading between the lines

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

  • If the bound extends to higher dimensions, it could resolve the full non-symmetric Mahler conjecture.
  • Computational checks on standard polytopes like the tetrahedron could verify the equality case.
  • Connections to symmetric cases might reveal how asymmetry affects the minimal product.

Load-bearing premise

The Santaló point exists and is unique for every convex body in R^3 and serves as the reference point that defines the non-symmetric volume product.

What would settle it

A convex body in R^3 whose non-symmetric volume product at the Santaló point is less than 64/9 would disprove the claim.

read the original abstract

We prove the non-symmetric Mahler conjecture in dimension three. More precisely, we prove the sharp lower bound \[ \mathcal P(K) \geq \frac{64}{9} \] for every convex body $K \subset \mathbb R^3$, where $\mathcal P(K)$ denotes the non-symmetric volume product with respect to the Santal\'o point.

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

1 major / 1 minor

Summary. The manuscript proves the non-symmetric Mahler conjecture in dimension three by establishing the sharp lower bound P(K) ≥ 64/9 for the non-symmetric volume product P(K) of every convex body K ⊂ R^3 with respect to its Santaló point.

Significance. If correct, the result resolves the three-dimensional case of a longstanding conjecture in convex geometry, confirming that the minimal non-symmetric volume product is 64/9 and is attained at a tetrahedron. This supplies a concrete sharp constant for an affine-invariant functional and may facilitate progress on related problems such as the symmetric Mahler conjecture or higher-dimensional analogs.

major comments (1)
  1. [reduction step from polytopes to general bodies] The reduction from general convex bodies to polytopes (implicit in the passage to the infimum) requires that the Santaló point s(K) is unique and that the map K ↦ vol(K^o_{s(K)}) is lower semi-continuous with respect to Hausdorff convergence. No explicit verification or citation establishing continuity of s(K) in the C^0 topology or lower semi-continuity of the polar volume appears in the argument, which is load-bearing for controlling the limit inferior and confirming that inf P(K) = 64/9 is attained.
minor comments (1)
  1. The abstract states the result but does not indicate the key technical steps (e.g., whether the proof proceeds by direct computation on tetrahedra followed by approximation). Adding one sentence on the strategy would improve readability.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript and for identifying the need to make the reduction step from polytopes to general convex bodies fully rigorous. We address the comment in detail below and will revise the manuscript to incorporate the requested clarifications.

read point-by-point responses

  1. Referee: [reduction step from polytopes to general bodies] The reduction from general convex bodies to polytopes (implicit in the passage to the infimum) requires that the Santaló point s(K) is unique and that the map K ↦ vol(K^o_{s(K)}) is lower semi-continuous with respect to Hausdorff convergence. No explicit verification or citation establishing continuity of s(K) in the C^0 topology or lower semi-continuity of the polar volume appears in the argument, which is load-bearing for controlling the limit inferior and confirming that inf P(K) = 64/9 is attained.

    Authors: We agree that the reduction step requires explicit justification of these continuity properties, which was not spelled out in the original submission. The uniqueness of the Santaló point s(K) is a standard fact in convex geometry: it is the unique minimizer of the volume of the polar body and follows from the strict convexity of the map x ↦ vol((K - x)^o) (see, e.g., Schneider, Convex Bodies: The Brunn–Minkowski Theory, Theorem 10.3.2, or the original work of Santaló). For the lower semi-continuity of K ↦ vol(K^o_{s(K)}), we will add a short lemma in the revised manuscript. The argument proceeds by first establishing continuity of the Santaló-point map K ↦ s(K) with respect to Hausdorff convergence on the space of convex bodies (this follows from the continuity of the volume functional and a compactness argument using Blaschke selection). Once s(K_n) → s(K), the Hausdorff convergence of the translated bodies K_n - s(K_n) to K - s(K) implies, via the relation between support functions and polar bodies, that liminf vol((K_n - s(K_n))^o) ≥ vol((K - s(K))^o). Combined with the continuity of vol(K) under Hausdorff convergence, this yields the desired lower semi-continuity of the non-symmetric volume product. We will include either a self-contained sketch or a precise citation to the relevant continuity results in the literature. This addition will make the passage to the infimum over all convex bodies fully rigorous. revision: yes

Circularity Check

0 steps flagged

Direct proof of non-symmetric Mahler bound with standard Santaló-point properties; no definitional or fitted-input circularity

full rationale

The manuscript states a direct proof that the non-symmetric volume product satisfies P(K) ≥ 64/9 for all convex bodies K in R^3, with the Santaló point serving as the reference. This bound is not obtained by fitting a parameter to a subset of data and then relabeling the fit as a prediction, nor is the target quantity defined in terms of itself. The existence and uniqueness of the Santaló point are standard facts in convex geometry and are not derived from the present result or from a self-citation chain that would render the central inequality tautological. Reduction steps from polytopes to general bodies rely on continuity properties that are external to the paper's own equations; no equation is shown to equal its own input by construction. The derivation therefore remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the existence of the Santaló point and on standard facts from convex geometry that are not re-derived in the abstract.

axioms (1)
  • domain assumption Existence and uniqueness of the Santaló point for every convex body in R^3
    The non-symmetric volume product is defined relative to this point; the abstract presupposes it exists and can be used as the reference.

pith-pipeline@v0.9.0 · 5578 in / 1164 out tokens · 36814 ms · 2026-05-22T10:46:48.042521+00:00 · methodology

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. The Symmetric Mahler Inequality in Dimension Three via Admissible Shadow Systems

    math.MG 2026-05 unverdicted novelty 6.0

    A new proof shows that every origin-symmetric convex body K in R^3 satisfies |K| |K^o| >= 32/3 via symmetric admissible shadow systems.

Reference graph

Works this paper leans on

22 extracted references · 22 canonical work pages · cited by 1 Pith paper

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