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LA Probability Forum

Thursday, June 2, 2022
6:00pm to 7:00pm
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Linde Hall 310
Convex Cylinders and the Symmetric Gaussian Isoperimetric Problem
Steven Heilman, Mathematics Department, USC,

The symmetric Gaussian isoperimetric problem of Barthe (2001) asks for the symmetric Euclidean set of fixed Gaussian volume with smallest Gaussian surface area. We show that, if such a set is a cylinder, then this set or its complement must be convex. Moreover, except for one case for the sign of the Gaussian mean curvature of the optimal set in R^n, the boundary of the optimal set must be r S^k × R^(n-k) for some (n-1)^(1/2) ≤ r ≤ (n+1)^(1/2) and for some 1 ≤ k ≤ n-1 (if k=0 we do not constrain r). It has been known since the 1970s that a Euclidean set with fixed Gaussian volume and smallest Gaussian surface area is a half space, and this implies e.g. Gaussian concentration of measure. The added symmetry assumption of Barthe's problem introduces extra difficulties.

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