Symmetric convex sets with minimal Gaussian surface area

Symmetric Convex Sets with Minimal Gaussian Surface Area

Let Ω ⊆ R have minimal Gaussian surface area among all sets satisfying Ω = −Ω with fixed Gaussian volume. Let A = Ax be the second fundamental form of ∂Ω at x, i.e. A is the matrix of first order partial derivatives of the unit normal vector at x ∈ ∂Ω. For any x = (x1, . . . , xn+1) ∈ R, let γn(x) = (2π)−n/2e−(x 2 1+···+x 2 n+1. Let ‖A‖ be the sum of the squares of the entries of A, and let ‖A‖...

Symmetric Convex Sets with Minimal Gaussian Surface Area

Abstract. Let Ω ⊆ R have minimal Gaussian surface area among all sets satisfying Ω = −Ω with fixed Gaussian volume. Let A = Ax be the second fundamental form of ∂Ω at x, i.e. A is the matrix of first order partial derivatives of the unit normal vector at x ∈ ∂Ω. For any x = (x1, . . . , xn+1) ∈ R, let γn(x) = (2π)−n/2e 2 1+···+x 2 n+1. Let ‖A‖2 be the sum of the squares of the entries of A, and...

Gaussian Correlation Conjecture for Symmetric Convex Sets

Gaussian correlation conjecture states that the Gaussian measure of the intersection of two symmetric convex sets is greater or equal to the product of the measures. In this paper, firstly we prove that the inequality holds when one of the two convex sets is the intersection of finite centered ellipsoids and the other one is simply symmetric. Then we prove that any symmetric convex set can be a...

Gaussian measures of dilatations of convex symmetric sets

Convex Sets and Minimal Sublinear Functions

We show that, given a closed convex set K with the origin in its interior, the support function of the set {y ∈ K∗ | ∃x ∈ K such that xy = 1} is the pointwise smallest sublinear function σ such that K = {x |σ(x) ≤ 1}.