{
  "id": "symplectic-balls-and-the-symmetric-mahler-inequality",
  "title": "Symplectic balls and the symmetric Mahler inequality",
  "description": "OpenAI's theorem (2026) that for every origin-symmetric convex body K in R^n, n >= 2, every symplectic ball of capacity c < 4 embeds into int K x int K°, and its consequence, the symmetric Mahler inequality vol(K) vol(K°) >= 4^n/n!. Polar bodies, support functions and strip bodies; one- and two-forms, the Pfaffian and volume preservation, flows and Moser's method; plurisubharmonic potentials and dd^c; a ball of capacity below pi k from a holomorphic zero of order k; a planar conformal map from the arctangent series and a uniform estimate for its horizontal slice integrals; a symplectic realization inside int K x S int K° with S close to pi k/4, rescaling, and polar approximation of general bodies.",
  "units": [
    {
      "id": "polar-bodies-and-the-volume-product",
      "title": "Polar bodies and the volume product",
      "reader": "polar-bodies-and-the-volume-product.html",
      "source": "src/polar-bodies-and-the-volume-product.md",
      "source_sha256": "287983bc07dd14806318c7bdd7d186566eaf9ca725485f0e1ba43a4abb50dae9",
      "status": "author_self_checked"
    },
    {
      "id": "forms-flows-and-moser-s-method",
      "title": "Forms, flows and Moser's method",
      "reader": "forms-flows-and-moser-s-method.html",
      "source": "src/forms-flows-and-moser-s-method.md",
      "source_sha256": "6d62c0edd67e687da5ada59ff1d68c38bb7cb7ff4170b861240a70f9c063d4a7",
      "status": "author_self_checked"
    },
    {
      "id": "plurisubharmonic-potentials",
      "title": "Plurisubharmonic potentials",
      "reader": "plurisubharmonic-potentials.html",
      "source": "src/plurisubharmonic-potentials.md",
      "source_sha256": "92d022818a272e776d5375677733917c323c1fcdb5aed84f603b7577706fd708",
      "status": "author_self_checked"
    },
    {
      "id": "a-ball-from-a-holomorphic-zero",
      "title": "A ball from a holomorphic zero",
      "reader": "a-ball-from-a-holomorphic-zero.html",
      "source": "src/a-ball-from-a-holomorphic-zero.md",
      "source_sha256": "ffaae3cc5834d47e4a556dd650ccc0ebe300629d672f142f77795966f83eaf26",
      "status": "author_self_checked"
    },
    {
      "id": "a-planar-conformal-map",
      "title": "A planar conformal map",
      "reader": "a-planar-conformal-map.html",
      "source": "src/a-planar-conformal-map.md",
      "source_sha256": "b2ce1ea3a3cd025e78580341125e6afdedd4d8a4c23fa9089437da0ebc81ae06",
      "status": "author_self_checked"
    },
    {
      "id": "the-uniform-slice-estimate",
      "title": "The uniform slice estimate",
      "reader": "the-uniform-slice-estimate.html",
      "source": "src/the-uniform-slice-estimate.md",
      "source_sha256": "d8925f2e1b4fdbfc9f4d4e5c788d823cec647ff11b7b149e77a353e040007a2a",
      "status": "author_self_checked"
    },
    {
      "id": "realization-in-a-polar-product",
      "title": "Realization in a polar product",
      "reader": "realization-in-a-polar-product.html",
      "source": "src/realization-in-a-polar-product.md",
      "source_sha256": "adcd3585dae39e934463c147292afc5e131df289db3c48c642a81804716219e0",
      "status": "author_self_checked"
    }
  ],
  "prerequisites": [
    "linear algebra and multivariable calculus, with the inverse and implicit function theorems (core course Real Analysis II, Lebl, Theorems 8.5.1 and 8.5.6)",
    "complex analysis of one variable (core course Complex Analysis, Lebl's guide, cited by number)",
    "Lebesgue measure, dominated convergence, differentiation under the integral sign, Fubini's theorem and the change of variables formula (core course Measure and Integration, Fremlin, cited by number)",
    "local solutions of ordinary differential equations and their smooth dependence on initial data (programme lesson Local tools for bundles and transport, Theorem 2.1; inverse function theorem, Theorem 1.2)"
  ],
  "status": "author_self_checked",
  "proof_providers": "providers.json",
  "provenance": "provenance.json",
  "licence": "CC0-1.0"
}