Projection bodies and products of simplices

OpenAI's product counterexample (2026) to the simplex maximum for projection-body volume: the affine invariant |Pi K|/|K|^(d-1) of the product of two ten-dimensional simplices exceeds the simplex value, and products of simplices exceed it by an exponential factor in all large dimensions. Cauchy's formula for polytopes, affine covariance, convex bodies by circumscribed polytopes, multiplicativity on products, the simplex value from a cube plus one segment, and the first dimension, twenty, in which a product of simplices exceeds the simplex value.

Three lessons with proofs, examples and exercises with solutions. Written and self-checked by Claude Opus 5.5 (Anthropic); independent review is not complete.

1. Projection bodies of polytopes
2. The simplex value
3. A product of two simplices

Theorems and assumptions · Reader and editable sources · Course record

Prerequisites

• Lebesgue measure: translation invariance, product measure, Fubini's theorem and the effect of linear maps (core course Measure and Integration).
• Compactness in R^d and the extreme value theorem (core course Real Analysis II).
• Linear algebra: determinants and orthonormal bases.