{
  "id": "the-logarithmic-brunn-minkowski-inequality",
  "title": "The logarithmic Brunn-Minkowski inequality",
  "description": "Support functions and Wulff bodies; the logarithmic Brunn-Minkowski inequality for origin-symmetric convex bodies and its reduction to a variance bound for soft polytopes; the additive inequality for powers between 0 and 1. Moment pairs; adjoint identities; the operator A and its test functions. The tensor estimate, with the difference written as a sum of squares. Density of the even test functions. The variance bound for soft polytopes and the logarithmic Brunn-Minkowski inequality, from two named hypotheses; the Gaussian case without hypotheses. The course is an exposition of OpenAI's proof (2026) of the logarithmic Brunn-Minkowski inequality; it proves the inequality from two statements of the literature that it names as hypotheses.",
  "units": [
    {
      "id": "wulff-bodies-and-the-reduction-to-a-variance-bound",
      "title": "Wulff bodies and the reduction to a variance bound",
      "reader": "wulff-bodies-and-the-reduction-to-a-variance-bound.html",
      "source": "src/wulff-bodies-and-the-reduction-to-a-variance-bound.md",
      "source_sha256": "fe927b51b9414afc736c2ce1a0445b43bf9724c9665c46578970df61d30addc3",
      "status": "author_self_checked"
    },
    {
      "id": "moment-coordinates-and-their-adjoints",
      "title": "Moment coordinates and their adjoints",
      "reader": "moment-coordinates-and-their-adjoints.html",
      "source": "src/moment-coordinates-and-their-adjoints.md",
      "source_sha256": "1bcd631f2eb786aa72780a55de6b33597da66fe0ea39a8a0c83406d96a51c1bb",
      "status": "author_self_checked"
    },
    {
      "id": "the-tensor-estimate",
      "title": "The tensor estimate",
      "reader": "the-tensor-estimate.html",
      "source": "src/the-tensor-estimate.md",
      "source_sha256": "7eabe7bc521e684c53615dc5400a65283c550f9116169be293357687db44351a",
      "status": "author_self_checked"
    },
    {
      "id": "density-of-the-even-tests",
      "title": "Density of the even tests",
      "reader": "density-of-the-even-tests.html",
      "source": "src/density-of-the-even-tests.md",
      "source_sha256": "297d898b7e930acd256ef07bad4c0f2ccbc7246d21e1a9bda75d7714eeb4524a",
      "status": "author_self_checked"
    },
    {
      "id": "the-variance-bound-and-the-inequality",
      "title": "The variance bound and the inequality",
      "reader": "the-variance-bound-and-the-inequality.html",
      "source": "src/the-variance-bound-and-the-inequality.md",
      "source_sha256": "835a43830a379f7fad1a2cf729223308c9aee31d9842280f4415d9d01f477d74",
      "status": "author_self_checked"
    }
  ],
  "prerequisites": [
    "the Lebesgue integral in R^n with the theorems of dominated and monotone convergence (core course Measure and Integration)",
    "compact convex sets in R^n and Hoelder's inequality for finite sums; the lesson proves the facts about support functions that it uses"
  ],
  "status": "author_self_checked",
  "proof_providers": "providers.json",
  "provenance": "provenance.json",
  "licence": "CC0-1.0"
}