1 · The essentials
These results concern how the size of a solid convex shape relates to the sizes of its shadows in different directions. Those shadow sizes are recorded by another shape, called its projection body. The main result says that, in every dimension at least four, ellipsoids have the smallest projection-body volume among convex shapes of the same volume. Only ellipsoids attain that smallest value. A companion result challenges a proposed answer at the opposite end: that a simplex, the higher-dimensional counterpart of a triangle or tetrahedron, gives the largest value after adjusting for size. It supplies a twenty-dimensional counterexample built from two ten-dimensional simplices. Both manuscripts are listed as Lean-formalized.
A picture to help (an imperfect image, not to be taken literally)
As an imperfect image, imagine comparing objects made from the same amount of material by collecting their silhouettes from every direction. The question is not which object has the smallest single silhouette, but which makes the smallest combined geometric record of all those silhouettes.
Why it matters
The main result settles Petty’s projection-volume conjecture in the remaining dimensions identified by the repository summary, namely dimensions at least four. The companion result shows that the proposed simplex maximum does not hold in general. Together, they give a sharp answer to the minimum question while ruling out a proposed answer to the maximum question.
2 · Going further
For a convex body , its projection body records the volumes of its projections in different directions. The main optimization problem fixes the volume of and asks which bodies minimize the volume of that projection body. Manuscript 1 establishes that, for , the minimizers are precisely ellipsoids. The repository describes this as resolving the remaining dimensions of Petty’s projection-volume conjecture and also reports the full Lutwak–Petty projection inequalities, without stating their formulas or hypotheses.
The companion manuscript concerns normalized projection-body volume and Brannen’s proposed simplex maximum. Its stated construction is the product of two ten-dimensional simplices: this has greater normalized projection-body volume than a twenty-dimensional simplex. Thus the product construction provides a counterexample rather than a classification of maximizers. The repository summary additionally reports that products of simplices exceed the proposed simplex maximum by an exponential factor in every sufficiently large dimension.
The abstracts give the conclusions but not the argument for the minimum theorem or the calculations behind the product comparison. Consequently, the supplied text supports describing the extremal shapes and the counterexample construction, but not reconstructing their proofs. Both manuscripts are marked Lean-formalized; the broader repository claims are not accompanied here by a detailed formalization scope.
Useful notions
- Convex shapes and convex bodies
- Volume in higher dimensions
- Projections and shadows
- Ellipsoids and simplices
- Minimization and maximization
What was known before / what is new
Before
The source identifies the minimum question as Petty’s projection-volume conjecture and describes dimensions at least four as the remaining dimensions. It also identifies a proposed simplex maximum due to Brannen. No further history or earlier proof methods are supplied.
New
The main Lean-formalized manuscript establishes the unique minimizing role of ellipsoids among convex bodies of fixed volume in every dimension at least four. The companion Lean-formalized manuscript supplies a counterexample using a product of two ten-dimensional simplices. Beyond those abstract statements, the repository reports the full Lutwak–Petty projection inequalities and an exponential advantage for products of simplices in every sufficiently large dimension.
Limits and assumptions
The minimum theorem stated here concerns convex bodies of fixed volume and dimensions ; the supplied abstract does not state results for lower dimensions. The companion abstract gives a specific twenty-dimensional counterexample, not a description of the actual maximizers. Both manuscripts are marked Lean-formalized, but the supplied text does not specify whether the additional repository claims about the full Lutwak–Petty inequalities and exponential separation are included in that formalization. It also gives neither the normalization formula nor a threshold for “sufficiently large” dimensions.
Mini glossary
- Convex body —
- A solid, bounded convex shape that contains the line segment joining any two of its points.
- Projection body —
- A body encoding the volumes of another convex body’s projections in different directions.
- Ellipsoid —
- A higher-dimensional counterpart of an ellipse, obtained by stretching a ball along its axes.
- Simplex —
- The higher-dimensional counterpart of a triangle or tetrahedron.
- Normalized projection-body volume —
- Projection-body volume adjusted to compare bodies independently of their overall size.
- Lean formalization —
- A mathematical statement and its proof expressed for verification in the Lean proof assistant.
Keywords : Projection bodies · Petty’s conjecture · Simplex counterexample
AI-generated explanation based on the manuscript. Unofficial. Refer to the manuscript for the authoritative version.
3 · Technical details and sources
Claim made in the manuscript (source text)
Proves Petty's projection-volume conjecture in the remaining dimensions n ≥ 4: ellipsoids uniquely minimize projection-body volume at fixed body volume. Also establishes the full Lutwak–Petty projection inequalities. In contrast, products of simplices exceed Brannen's proposed simplex maximum for normalized projection-body volume by an exponential factor in every sufficiently large dimension.
Verified by Lean: the Lean software checked the part of the reasoning described below, sometimes only part of the manuscript. This does not mean independent mathematicians have validated the result.
Manuscripts in this family
Petty’s projection-volume conjecture in dimensions at least four
24/09/2026We prove that ellipsoids uniquely minimize the volume of the projection body among convex bodies of fixed volume in every dimension at least four. This proves Petty's projection-volume conjecture in these dimensions.
A product counterexample to the simplex maximum for projection-body volume
24/09/2026The product of two ten-dimensional simplices has larger normalized projection-body volume than a twenty-dimensional simplex. This gives a counterexample to Brannen's proposed simplex maximum.