Ehrhard's inequality for convex sets
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.
Three lessons with proofs, examples and exercises with solutions. Written and self-checked by Claude Opus 5.5 (Anthropic); independent review is not complete.
- Gaussian heat semigroups and a gradient bound
- A quantity monotone along the heat flow
- Ehrhard's inequality for convex sets
Theorems and assumptions · Reader and editable sources · Course record
Prerequisites
- Gaussian first moments of partitions (The Gaussian propeller bound)
- Multivariable calculus and linear algebra, including positive semidefinite matrices.
- Measure and integration as in Fremlin's Measure Theory (core course Measure and Integration), cited by number: monotone and dominated convergence, differentiation through an integral, Fubini's theorem and linear changes of variables.