Isoperimetry and the Ornstein-uhlenbeck Operator

We first survey the relation between the classical isoperimetric problem, the isoperimetric problem for the Gaussian measure, and the Ornstein-Uhlenbeck operator. We then describe a generalization of these results, which was posed by Isaksson and Mossel [10]. Some results on the conjecture of Isaksson and Mossel [7] will then be described. Both probabilistic and analytic methods will be emphasized. Finally, we describe applications to theoretical computer science. 1. Euclidean Isoperimetry Let n ≥ 1 be an integer, let A ⊆ R be a Borel set with smooth boundary ∂A. Let voln(A) denote the Euclidean volume of A, and let voln−1(∂A) denote the Euclidean area of the boundary of A. Theorem 1.1 (Euclidean Isoperimetric Inequality). The Euclidean ball has the smallest boundary among all sets of fixed volume. That is, let B ⊆ R be a Euclidean ball such that voln(A) = voln(B). Then voln−1(∂A) ≥ voln−1(∂B). One way of proving this theorem uses symmetrization. That is, given any A ⊆ R, we rearrange A into a more symmetric set with smaller surface area but identical volume. If A ⊆ R, we take any line L that passes through A, and then write A as a union of onedimensional sets that are perpendicular to L. Then, slide each such one-dimensional set such that it is a line segment symmetric with respect to reflection across L. The rearranged set has the same volume as A. Intuitively, the rearranged set also has smaller surface area, since the surface area at a point on the boundary of A becomes averaged after rearrangement.