1. An Isoperimetric Result on High-Dimensional Spheres
- Author
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Barnes, Leighton Pate, Ozgur, Ayfer, and Wu, Xiugang
- Subjects
FOS: Computer and information sciences ,Mathematics - Metric Geometry ,Mathematics - Classical Analysis and ODEs ,Computer Science - Information Theory ,Information Theory (cs.IT) ,Probability (math.PR) ,Classical Analysis and ODEs (math.CA) ,FOS: Mathematics ,Metric Geometry (math.MG) ,Mathematics - Probability - Abstract
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 $\mathbb{S}^{m-1}$, and let $\mathbf{y}\in \mathbb{S}^{m-1}$ be a randomly chosen point on the sphere. What is the measure of the intersection of the $t$-neighborhood of the point $\mathbf{y}$ 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., Comment: arXiv admin note: text overlap with arXiv:1701.02043
- Published
- 2018
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