An Isoperimetric Result on High-Dimensional Spheres

11/20/2018

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by   Leighton Pate Barnes, et al.

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We consider an extremal problem for subsets of high-dimensional spheres that can be thought of as an extension of the classical isoperimetric problem on the sphere. Let A be a subset of the (m-1)-dimensional sphere S^m-1, and let y∈S^m-1 be a randomly chosen point on the sphere. What is the measure of the intersection of the t-neighborhood of the point y with the subset A? We show that with high probability this intersection is approximately as large as the intersection that would occur with high probability if A were a spherical cap of the same measure.