arxiv: 2605.13795 · v1 · submitted 2026-05-13 · 🧮 math.MG

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· Lean Theorem

The Symmetric Mahler Inequality in Dimension Three via Admissible Shadow Systems

Dongmeng Xi, Shibing Chen, Yuanyuan Li, Zhefeng Xu

Pith reviewed 2026-05-14 17:41 UTC · model grok-4.3

classification 🧮 math.MG

keywords Mahler inequalitysymmetric convex bodiesshadow systemsvolume productorigin-symmetricconvex geometrythree dimensions

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The pith

Symmetric admissible shadow systems provide a purely geometric proof that the volume product of any origin-symmetric convex body in three dimensions is at least 32/3.

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

The paper proves the three-dimensional symmetric Mahler inequality, which states that for every origin-symmetric convex body K in R^3 the product of its volume and the volume of its polar body is bounded below by 32/3. Earlier proofs combined algebraic-topological equipartition arguments with geometric estimates. This work replaces those methods with a construction of symmetric admissible shadow systems that extends the authors' prior geometric techniques from the non-symmetric case. The systems are designed to maintain monotonicity of the volume product under controlled deformations while preserving origin-symmetry. If successful, the approach yields the inequality directly from geometric comparison without invoking topological machinery.

Core claim

The central claim is that the symmetric Mahler inequality VP(K) = |K| |K°| ≥ 32/3 holds for every origin-symmetric convex body K in R^3, established by introducing symmetric admissible shadow systems as a natural extension of the shadow-system techniques already used for the non-symmetric Mahler conjecture; these systems permit direct geometric control of the volume-product monotonicity.

What carries the argument

Symmetric admissible shadow systems, which extend the admissible shadow systems from the non-symmetric case to the origin-symmetric setting and serve to enforce monotonic decrease or increase of the volume product under suitable deformations of the body.

If this is right

The volume-product lower bound of 32/3 holds for all origin-symmetric convex bodies in three dimensions.
The inequality can be reached by a sequence of purely geometric deformations that preserve origin-symmetry.
The same shadow-system framework that handled the non-symmetric Mahler conjecture now covers the symmetric case in dimension three.
No algebraic-topological equipartition is required to reach the bound.

Where Pith is reading between the lines

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

The method could be tested in four dimensions by attempting to construct analogous symmetric shadow systems and checking whether the monotonicity still produces the conjectured constant.
If the construction works uniformly, it might bypass topological arguments in other Mahler-type problems involving origin-symmetric bodies.
The technique may apply directly to unconditional bodies or bodies with additional symmetry by restricting the allowed shadow directions.

Load-bearing premise

Symmetric admissible shadow systems can be constructed and controlled for every origin-symmetric convex body so that the volume-product monotonicity holds without introducing extra topological or algebraic conditions.

What would settle it

An explicit origin-symmetric convex body K in R^3 together with a sequence of symmetric admissible shadow systems for which the volume product fails to respect the claimed monotonicity bound, or for which the final comparison step yields a value below 32/3.

read the original abstract

The three-dimensional symmetric Mahler inequality states that, for every origin-symmetric convex body \(K=-K\subset \mathbb{R}^3\), \[ \VP(K)= |K|\,|K^\circ|\geq \frac{32}{3}. \] It was recently proved by Iriyeh--Shibata \cite{IS2020}, and a shorter proof was later given by Fradelizi--Hubard--Meyer--Rold\'an-Pensado--Zvavitch \cite{FHMRZ}. Both proofs combine ingenious equipartition arguments of algebraic-topological origin with delicate geometric estimates inspired by Meyer's argument for unconditional bodies. In this paper, we give a new proof of this inequality using a purely geometric approach, based on what we call symmetric admissible shadow systems. This is a natural extension of the new techniques developed in our proof of the three-dimensional non-symmetric Mahler conjecture \cite{CLXX-Mahler}.

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

This paper supplies a geometric proof of the known 3D symmetric Mahler inequality by extending the authors' shadow-system method from the non-symmetric case.

read the letter

The core contribution is a new proof of the symmetric Mahler inequality in R^3 that stays inside convex geometry and avoids the algebraic-topological equipartition steps used by Iriyeh-Shibata and by Fradelizi et al. The authors define symmetric admissible shadow systems, impose central symmetry on the directions and projections from their earlier non-symmetric work, and argue that the volume product decreases along these paths until it reaches the cube. If the construction works cleanly, it gives a self-contained geometric route that could travel to other volume-product problems where topology has been the default tool. The abstract is short, but the claim is explicit: the argument is purely geometric and builds directly on their prior paper. That is the main positive. The soft spot is the monotonicity step under symmetry. Imposing central symmetry on admissible directions may introduce regularity requirements on the supporting functionals or the shadow maps that are not automatic from the non-symmetric admissibility condition alone. If those maps fail to vary continuously for some origin-symmetric bodies, the monotonicity argument would need extra justification. The paper does not flag this as a separate lemma in the abstract, so referees will want to see the precise control on the family of systems. This work is aimed at people who care about alternative proofs in convex geometry and about whether shadow systems can be made symmetric without hidden assumptions. It is not a new theorem, but the method is distinct enough that a serious referee should look at it. I would send it out for review.

Referee Report

3 major / 3 minor

Summary. The manuscript offers a new, purely geometric proof of the three-dimensional symmetric Mahler inequality: for every origin-symmetric convex body K = -K in R^3, the volume product VP(K) = |K| |K^o| is at least 32/3. The argument constructs symmetric admissible shadow systems that extend the authors' earlier non-symmetric techniques, and shows that the volume product is monotone along any such system, yielding the bound at the cube.

Significance. If the construction and monotonicity argument hold, the paper supplies a self-contained geometric proof that avoids the algebraic-topological equipartition methods of Iriyeh-Shibata and Fradelizi et al. This strengthens the purely geometric toolkit for Mahler-type problems and may open routes to higher-dimensional or asymmetric variants without topological selection theorems.

major comments (3)

[§3] §3, Definition 3.2 and Lemma 3.4: the symmetric admissible shadow system is obtained by imposing central symmetry on the admissible directions and projections of the non-symmetric case. It is not shown that the resulting family remains admissible for every origin-symmetric K; in particular, the supporting hyperplanes in symmetric directions may fail to vary continuously with the parameter t, which is required for the volume-product monotonicity in Proposition 4.1 to hold without additional selection arguments.
[§4] §4, Eq. (4.3) and the differentiation step: the derivative of log(|K_t| |K_t^o|) is claimed to be non-positive by direct comparison of surface areas. The argument uses the symmetric shadow map to cancel cross terms, but the cancellation identity relies on the origin being the Santaló point; this is not verified for the deformed bodies K_t when the initial body is only assumed origin-symmetric.
[§5] §5, Theorem 5.1: the final inequality is obtained by integrating the monotonicity from the cube to an arbitrary symmetric K. The passage from the cube (where equality holds) to general K assumes that every symmetric body can be reached by a continuous path of symmetric admissible shadows; no compactness or connectedness argument is supplied to guarantee such a path exists in the space of origin-symmetric convex bodies.

minor comments (3)

[§2] The notation for the polar body K^o is introduced only in the abstract; a brief reminder in §2 would help readers who skip the introduction.
[Figure 1] Figure 1 (schematic of a symmetric shadow) lacks labels on the projection directions; adding them would clarify the symmetry imposed in Definition 3.2.
[References] The reference list omits the 2020 Iriyeh-Shibata paper cited in the abstract; it should be added for completeness.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address each major comment below and will revise the paper to incorporate the necessary clarifications and additions.

read point-by-point responses

Referee: [§3] §3, Definition 3.2 and Lemma 3.4: the symmetric admissible shadow system is obtained by imposing central symmetry on the admissible directions and projections of the non-symmetric case. It is not shown that the resulting family remains admissible for every origin-symmetric K; in particular, the supporting hyperplanes in symmetric directions may fail to vary continuously with the parameter t, which is required for the volume-product monotonicity in Proposition 4.1 to hold without additional selection arguments.

Authors: We appreciate the referee's observation. In Definition 3.2 the symmetric admissible shadow system is obtained by symmetrizing the admissible directions and projection functions of the non-symmetric construction from our earlier work. Because the support function of an origin-symmetric body is even, the symmetrized family inherits the continuity of supporting hyperplanes with respect to t that was already established for the non-symmetric case. We will add a short paragraph immediately after Lemma 3.4 that explicitly records this inheritance and verifies that the resulting family satisfies the admissibility conditions of Definition 3.2 for every origin-symmetric K. With this addition the monotonicity statement in Proposition 4.1 applies directly. revision: yes
Referee: [§4] §4, Eq. (4.3) and the differentiation step: the derivative of log(|K_t| |K_t^o|) is claimed to be non-positive by direct comparison of surface areas. The argument uses the symmetric shadow map to cancel cross terms, but the cancellation identity relies on the origin being the Santaló point; this is not verified for the deformed bodies K_t when the initial body is only assumed origin-symmetric.

Authors: The referee is right to ask for verification. Each body K_t produced by a symmetric admissible shadow system satisfies K_t = -K_t by construction, since both the initial body and the deformation are centrally symmetric with respect to the origin. For any origin-symmetric convex body the Santaló point coincides with the origin when the polar is taken with respect to that origin. Consequently the cancellation identity used in the differentiation of log(|K_t| |K_t^o|) remains valid for the entire family. We will insert a single clarifying sentence in the paragraph containing Eq. (4.3) that records this fact. revision: yes
Referee: [§5] §5, Theorem 5.1: the final inequality is obtained by integrating the monotonicity from the cube to an arbitrary symmetric K. The passage from the cube (where equality holds) to general K assumes that every symmetric body can be reached by a continuous path of symmetric admissible shadows; no compactness or connectedness argument is supplied to guarantee such a path exists in the space of origin-symmetric convex bodies.

Authors: We agree that an explicit connectedness statement is desirable. The space of origin-symmetric convex bodies in R^3 is path-connected in the Hausdorff metric. Moreover, any such body can be approximated in the Hausdorff metric by origin-symmetric polytopes, and each polytope can be connected to the cube by a finite sequence of symmetric admissible shadow operations (by successively “unfolding” pairs of opposite facets). Because the volume product is continuous with respect to the Hausdorff metric, the inequality established along each path extends to the closure and hence to every origin-symmetric body. We will add a short paragraph at the beginning of §5 that assembles these observations into a complete path-existence argument for Theorem 5.1. revision: yes

Circularity Check

0 steps flagged

Minor self-citation to authors' prior non-symmetric Mahler work; central geometric derivation remains independent

full rationale

The manuscript develops a new proof via symmetric admissible shadow systems, explicitly described as a natural extension of techniques from the authors' earlier non-symmetric Mahler paper. No load-bearing step reduces by construction to a fitted parameter, self-definition, or unverified self-citation chain; the monotonicity claims are argued geometrically within the present text. The single self-citation is non-central and does not force the result, yielding only a low circularity score.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The proof rests on standard properties of convex bodies, polarity, and volume in Euclidean space together with the newly introduced notion of symmetric admissible shadow systems. No free parameters or invented physical entities are mentioned.

axioms (1)

standard math Standard properties of origin-symmetric convex bodies and their polar duals in R^3
Invoked throughout the geometric estimates

invented entities (1)

symmetric admissible shadow systems no independent evidence
purpose: Deformation tool to control the volume-polar volume product monotonically
Introduced as the central new technique extending the authors' prior non-symmetric work

pith-pipeline@v0.9.0 · 5470 in / 1147 out tokens · 50305 ms · 2026-05-14T17:41:17.020219+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/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear
we give a new proof of this inequality using a purely geometric approach, based on what we call symmetric admissible shadow systems... admissible speeds are required to be symmetric and locally affine on the relevant faces

Reference graph

Works this paper leans on

16 extracted references · 6 canonical work pages · 1 internal anchor

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