{
  "id": "ehrhards-inequality-for-convex-sets",
  "title": "Ehrhard's inequality for convex sets",
  "description": "A complete proof of Ehrhard's inequality for closed convex sets: Phi^{-1} of the Gaussian measure of lambda K + (1 - lambda) L is at least lambda Phi^{-1}(gamma(K)) + (1 - lambda) Phi^{-1}(gamma(L)). Gaussian heat semigroups, the heat equation and a moving function under the semigroup; the sharp gradient bound |grad Q_t f| <= t^(-1/2) I(Q_t f) from a rearrangement inequality; the interpolation method of Neeman and Paouris with the heat semigroup: a quantity monotone along the flow of a pair of correlated Gaussian vectors; Lipschitz approximation of convex sets. Concavity along translates of a convex set follows.",
  "units": [
    {
      "id": "gaussian-heat-semigroups-and-a-gradient-bound",
      "title": "Gaussian heat semigroups and a gradient bound",
      "reader": "gaussian-heat-semigroups-and-a-gradient-bound.html",
      "source": "src/gaussian-heat-semigroups-and-a-gradient-bound.md",
      "source_sha256": "4691b7803160efe01f5f882ff7aeb04e51ef021ebc08034caaae9624c898d0fb",
      "status": "author_self_checked"
    },
    {
      "id": "a-quantity-monotone-along-the-heat-flow",
      "title": "A quantity monotone along the heat flow",
      "reader": "a-quantity-monotone-along-the-heat-flow.html",
      "source": "src/a-quantity-monotone-along-the-heat-flow.md",
      "source_sha256": "3611ea78703fbba38cbe8dc3c81815c15b99851ba773e0e975fbe2c1e6c549c8",
      "status": "author_self_checked"
    },
    {
      "id": "ehrhards-inequality-for-convex-sets",
      "title": "Ehrhard's inequality for convex sets",
      "reader": "ehrhards-inequality-for-convex-sets.html",
      "source": "src/ehrhards-inequality-for-convex-sets.md",
      "source_sha256": "6c1ac14f503be6960771bef7df47ae0b9cc5a4771c5421bfc4b4b1176c9ba282",
      "status": "author_self_checked"
    }
  ],
  "prerequisites": [
    "multivariable calculus and linear algebra (positive semidefinite matrices)",
    "measure and integration as in Fremlin's Measure Theory (core course Measure and Integration), cited by number: 123A, 123C, 123D, 252B, 263A"
  ],
  "status": "author_self_checked",
  "proof_providers": "providers.json",
  "provenance": "provenance.json",
  "licence": "CC0-1.0"
}