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Math-2416 · mathematics · Geometry

Petty's projection-body conjecture

Status Partially solved · unsolvedmath.comLLM status Open unverified

Statement

Definitions and mathematical statement

Shared notation.
$\mathbb N$, $\mathbb Z$, $\mathbb Q$, $\mathbb R$, and $\mathbb C$ denote the natural numbers, integers, rationals, reals, and complex numbers, respectively. All additional parameters and hypotheses are specified in the question.
Statement recorded in the supplied source.
For a convex body $K\subseteq\mathbb R^d$, $d\geq3$, let $\Pi K$ be its projection body. Is the affine-invariant ratio $|\Pi K|/|K|^{d-1}$ minimized exactly by ellipsoids? The support function of $\Pi K$ in a direction is the $(d-1)$-volume of the orthogonal projection of $K$ onto its perpendicular hyperplane.

Formulation references: [S1], [S2].

Short English statement

For a convex body $K\subseteq\mathbb R^d$, $d\geq3$, let $\Pi K$ be its projection body. Is the affine-invariant ratio $|\Pi K|/|K|^{d-1}$ minimized exactly by ellipsoids? The support function of $\Pi K$ in a direction is the $(d-1)$-volume of the orthogonal projection of $K$ onto its perpendicular hyperplane.

Sources

Catalogue entry

Statement licence: CC-BY-SA-4.0 · Source file

Contributions 2

Math-2416-A2 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2416

The linked Petty theorem begins in dimension four, whereas this statement also includes dimension three. Its ellipsoid minimum therefore covers the higher-dimensional clause without supplying the remaining case from this manuscript.

Petty’s Projection-Body Conjecture

OpenAI's claim

OpenAI presents the following claims relevant to this question, with manuscript numbers following paper order in Sources. In manuscript 1 (family 088; 2026-09-24), the asserted result is that ellipsoids uniquely minimize normalized projection-body volume among convex bodies in every dimension at least four.

Mathematical statement of the claimed result

Manuscript 1.
Let $\Pi K$ be the projection body, whose support function at unit $u$ equals the $(n-1)$-volume of the orthogonal projection of $K$, and let $\kappa_j$ be the volume of the Euclidean unit ball in $\mathbb R^j$. For every $n\ge4$ and every convex body $K\subset\mathbb R^n$, the manuscript claims \[ \frac{|\Pi K|}{|K|^{n-1}}\ge\kappa_{n-1}^{\,n}\kappa_n^{\,2-n}. \] Equality is claimed exactly for $K=a+TB_2^n$ with $T$ invertible. The dimension range starts at four; the theorem does not itself supply missing lower-dimensional statements.

Outline of the argument

For manuscript 1 (family 088; 2026-09-24), directional projection data are encoded in a measure associated with the body's gauge. A spherical harmonic sign estimate and a variational affine-position argument give a strict norm inequality, whose equality conditions identify ellipsoids.

Formal verification

For manuscript 1 (family 088; 2026-09-24), OpenAI lists Lean statements for this assertion: ellipsoids uniquely minimize normalized projection-body volume among convex bodies in every dimension at least four.

This coverage does not establish a formal proof of the complete problem as stated here.

The linked family Comparator checks are PettyProjectionVolume.lean, ProjectionCounterexample.lean, ProjectionVolume.lean.

Their named declarations include OAI.Paper092.product_counterexample, OAI.PettyProjection.petty_projection_volume, OAI.ProjectionCounterexample.universal_simplex_upper_bound_false.

This site has not executed the Lean code. Described statement coverage is not an independent proof check.

Status caveat

The manuscripts were produced by an internal OpenAI model. OpenAI cautions that results without formalization can contain mistakes. This site has not independently checked the proofs; the label records the scope of OpenAI's claim.

Sources

Go to contribution →

Math-2416-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2416

The linked Petty theorem begins in dimension four, whereas this statement also includes dimension three. Its ellipsoid minimum therefore covers the higher-dimensional clause without supplying the remaining case from this manuscript.

Petty's projection-body conjecture

OpenAI's claim

OpenAI presents the following claims relevant to this question, with manuscript numbers following paper order in Sources. In manuscript 1 (family 088; 2026-09-24), the asserted result is that ellipsoids uniquely minimize normalized projection-body volume among convex bodies in every dimension at least four.

Mathematical statement of the claimed result

Manuscript 1.
Let $\Pi K$ be the projection body, whose support function at unit $u$ equals the $(n-1)$-volume of the orthogonal projection of $K$, and let $\kappa_j$ be the volume of the Euclidean unit ball in $\mathbb R^j$. For every $n\ge4$ and every convex body $K\subset\mathbb R^n$, the manuscript claims \[ \frac{|\Pi K|}{|K|^{n-1}}\ge\kappa_{n-1}^{\,n}\kappa_n^{\,2-n}. \] Equality is claimed exactly for $K=a+TB_2^n$ with $T$ invertible. The dimension range starts at four; the theorem does not itself supply missing lower-dimensional statements.

Outline of the argument

For manuscript 1 (family 088; 2026-09-24), directional projection data are encoded in a measure associated with the body's gauge. A spherical harmonic sign estimate and a variational affine-position argument give a strict norm inequality, whose equality conditions identify ellipsoids.

Formal verification

For manuscript 1 (family 088; 2026-09-24), OpenAI lists Lean statements for this assertion: ellipsoids uniquely minimize normalized projection-body volume among convex bodies in every dimension at least four.

This coverage does not establish a formal proof of the complete problem as stated here.

The linked family Comparator checks are PettyProjectionVolume.lean, ProjectionCounterexample.lean, ProjectionVolume.lean.

Their named declarations include OAI.Paper092.product_counterexample, OAI.PettyProjection.petty_projection_volume, OAI.ProjectionCounterexample.universal_simplex_upper_bound_false.

This site has not executed the Lean code. Described statement coverage is not an independent proof check.

Status caveat

The manuscripts were produced by an internal OpenAI model. OpenAI cautions that results without formalization can contain mistakes. This site has not independently checked the proofs; the label records the scope of OpenAI's claim.

Sources

Go to contribution →