{
  "id": "projection-bodies-and-products-of-simplices",
  "title": "Projection bodies and products of simplices",
  "description": "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.",
  "units": [
    {
      "id": "projection-bodies-of-polytopes",
      "title": "Projection bodies of polytopes",
      "reader": "projection-bodies-of-polytopes.html",
      "source": "src/projection-bodies-of-polytopes.md",
      "source_sha256": "f5f36e8ef2f89eaee90d6352c00cc8b316d640188469f98f4d1101a749f1efcc",
      "status": "author_self_checked"
    },
    {
      "id": "the-simplex-value",
      "title": "The simplex value",
      "reader": "the-simplex-value.html",
      "source": "src/the-simplex-value.md",
      "source_sha256": "1b80008e39a46a5024cde5b7ddedd426dd2ed41bc769012da44d39cf000a0ace",
      "status": "author_self_checked"
    },
    {
      "id": "a-product-of-two-simplices",
      "title": "A product of two simplices",
      "reader": "a-product-of-two-simplices.html",
      "source": "src/a-product-of-two-simplices.md",
      "source_sha256": "20bf4bba90af7f898bb47fb40c1ae7cbcfc8074bba1dbc3f12414a81faa328ec",
      "status": "author_self_checked"
    }
  ],
  "prerequisites": [
    "Lebesgue measure: core course Measure and Integration (Fremlin, Measure Theory, Volumes 1 and 2): translation invariance (134A), Lebesgue measure on R^(r+s) as a product measure (251N), Fubini's theorem (252B), linear maps (263A)",
    "compactness in R^d and the extreme value theorem: core course Real Analysis II (Lebl, Basic Analysis, Theorem 7.5.6)",
    "linear algebra: determinants, orthonormal bases"
  ],
  "status": "author_self_checked",
  "proof_providers": "providers.json",
  "provenance": "provenance.json",
  "licence": "CC0-1.0"
}