Sharp projection-body inequalities and a counterexample to simplex maximization

What OpenAI claims

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. (Lean)

Papers (2)

1. Petty’s projection-volume conjecture in dimensions at least four

   24 September 2026 · Preprint, not peer reviewed · PDF

   We 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.

2. A product counterexample to the simplex maximum for projection-body volume

   24 September 2026 · Preprint, not peer reviewed · PDF

   The 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.

Status

Preprint. The papers have not been independently reviewed. Treat the result as OpenAI's claim until other mathematicians have checked it. If a video exists, it illustrates the statement; it is not the proof.