Convex geometry · affine invariants · polarity · extremal volume

Mahler Volume-Product Conjecture

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Which convex bodies minimize the affine-invariant product of their volume and the volume of their polar dual?

| K | | K^{\circ} | \geq \frac{(n + 1)^{n + 1}}{(n!)^{2}}; K = - K \Rightarrow | K | | K^{\circ} | \geq \frac{4^{n}}{n!}

Listed inAIM problem list: Mahler's conjecture and duality in convex geometryListed inWikipedia list of unsolved problems in mathematics

Known results and sources

A convex body and its polar exchange supporting directions and boundary geometry while their volume product remains unchanged by affine transformations.

Research problem

Exact mathematical statement

Let $K\subset\mathbb R^n$ be a convex body with nonempty interior. Translate its Santaló point $s(K)$ to the origin and let $K^\circ$ denote the polar of $K-s(K)$ . The general Mahler conjecture predicts

|K|\,|K^\circ|\ge \frac{(n+1)^{n+1}}{(n!)^2},

with conjectural equality precisely for simplices up to affine equivalence. If $K=-K$ , the centrally symmetric Mahler conjecture predicts

|K|\,|K^\circ|\ge \frac{4^n}{n!},

with equality conjectured for Hanner polytopes. Both statements are proved in dimension two; the symmetric statement is proved in dimension three. A recent preprint claims the general three-dimensional statement but has not been treated here as an established theorem. Both statements remain open in dimensions four and above.

Problem infographic

Problem at a glance

Scientific explainer for Mahler's volume-product conjectures showing an exact translated-ellipse, centered-ellipse, and reciprocal-axis polar-ellipse example, the affine-invariant product of volumes, the conjectured simplex and Hanner-polytope bounds, and the established versus open dimensional frontier. — Mahler's conjectures ask for the sharp lower volume product after Santaló centering: simplices in the general case and Hanner polytopes in the centrally symmetric case are the proposed extremizers.

Current mathematical picture

Where work on Mahler Volume-Product Conjecture stands

Recent proof claim under review

The current research keeps the general and centrally symmetric Mahler conjectures distinct. It records two principal active programs: a flag/KL-compensation route for arbitrary polytopes and a resonance-ray route restricted to centrally symmetric polytopes with at most 2n+2 vertices or facets. The current work reports exact reductions, four all-dimensional visibility chambers, structural consequences, counterexamples to several tempting shortcuts, and an exact ray census through m=6. The internal KL-compensation lemma and the full resonance-ray inequality remain open. Audit-sensitive derivations, standard external inputs, and the included Python census have not been independently checked or reproduced here. No formal proof is supplied; these remain source-reported results, not independently established evidence.

Leading routeFlag and internal KL-compensation route

Use the exact weighted maximal-flag lower bound, local equilibrium duals, max-path representation, canonical telescope, and endpoint Hall coupling; then prove that incompatibility among internal rank kernels forces enough aggregate KL gain to cover the partition deficit.

Route status · Active route

Main reductionLow-complexity symmetric class reaches resonance rays

The source reduces the 2n+2-vertex/facet symmetric class to a small-ball/Rademacher norm comparison and then to a discrete inequality on primitive positive resonance rays.

Evidence posture · Reported reduction

Completed special caseFour chambers and finite ray census reported

Four all-dimensional visibility chambers are recorded as packet-supported results; a separate exact census through m=6 is reported but remains unreproduced.

Evidence posture · Reported special case

Priority open bridgeAudit the finite-dimensional flag dual

Make the local equilibrium sets, boundary cases, minimax interchange, endpoint dual, canonical telescope, and support-variation Hall coupling fully rigorous.

Task status · Work already reported in progress

Research-record correctionResearch-record correction

We corrected the cited passages. We updated the highlighted open task or route. The mathematical claims and their status did not change.

Reader-facing record corrected; mathematics unchanged

Work mapped so far

Mahler Volume-Product Conjecture in numbers

1.5kretained lines of mathematical investigation1,456 in the current working snapshot

Argument development: 1,260 · 87%
Explored or eliminated routes: 6 · 0%
Computational analysis: 47 · 3%
Open obligations: 51 · 4%
Definitions and setup: 92 · 6%

11selected mapped statements5routes investigated6reported milestones7open questions6contribution-ready tasks

How this is measured

This measures retained mathematical investigation, not proximity to a proof. Code, data, logs, repeated text, operational instructions, and generated presentation copy are excluded.

Proposed byKurt MahlerProposed1939Status checkedAug 6, 2026

Source recordedAug 6, 2026

Recommended next task

Resolve the four-visible three-neighbor star

Prove the chamber inequality (30.2), derive a recursive reduction to the four proved chambers, or find an exact counterexample to the norm comparison or resonance-ray inequality.

Suggested move: Use master formulas (24A.4)–(24A.5), treat exceptional small dimensions separately, and distinguish neighbor-star from Boolean-subcube inclusion-exclusion.

What would count as progress

All exceptional small dimensions are treated rigorously.
The star inequality is proved, recursively reduced, or exactly refuted.
The next visibility down-set types are classified.

Argument map and routes

How the current approaches connect

Claims, reductions, open questions, active routes, and narrowed alternatives in one mathematical map.

Visible working map

Research route map

26 selected steps

Selected claims, active routes, useful failures, and open questions from the current research map. Arrows appear only for explicitly recorded relationships.

Scroll horizontally to explore the route

Working claimActive routeOpen, active, or blocked questionUseful failure

Working overview, not proof. The map shows selected recorded relationships; more nodes or edges do not establish correctness or completion.

Active routeFlag and internal KL-compensation route

Route status · Active route

Active routeLow-complexity symmetric resonance-ray route

Restrict to the 2n+2-vertex/facet symmetric class, use the small-ball/Rademacher normal form and chamber convexity, and prove the discrete inequality on every primitive positive extreme ray. Four visibility families are recorded as positive base chambers.

Route status · Active route

Active routeGeneral structural pyramid recursion

Try to force every general polytopal minimizer to be a pyramid using deletion inequalities, the pyramidal-label lemma, and non-symmetric flag compensation, then apply the exact Santaló-balanced pyramid recurrence and induct from dimension one.

Route status · Active route

Active routeFoundational and external-input audit

Independently check the high-risk finite-dimensional dual steps, support variation, Jacobians, chamber algebra, standard geometric inputs, and exact finite computations before any trust upgrade.

Route status · Active route

Explored alternatives

Other routes

1 recorded

Useful but insufficientDirectional shadow rigidity alone

The route remains useful as a necessary-condition and rejection tool, but it cannot classify Hanner bodies by itself because the source-reported P_n family passes the test while remaining non-Hanner.

Route status · Useful but insufficient

Route statements and reductions

Statements the next route can inspect and build on

Route statementGeneral Mahler volume-product conjecture

For every Santaló-centered convex body K in R^n, the volume product |K||K^circ| is at least (n+1)^(n+1)/(n!)^2, with conjectural equality precisely for simplices up to affine equivalence.