On the Isotropic Constant of Random Polytopes

Let X 1 , … , X N be independent random vectors uniformly distributed on an isotropic convex body K ⊂ R n , and let K N be the symmetric convex hull of X i ’s. We show that with high probability L K N ≤ C log ( 2 N / n ) , where C is an absolute constant. This result closes the gap in known estimate...

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Bibliographic Details
Published in:The Journal of geometric analysis 2016-01, Vol.26 (1), p.645-662
Main Authors: Alonso-Gutiérrez, David, Litvak, Alexander E., Tomczak-Jaegermann, Nicole
Format: Article
Language:English
Subjects:
Abstract Harmonic Analysis
Convex and Discrete Geometry
Differential Geometry
Dynamical Systems and Ergodic Theory
Fourier Analysis
Global Analysis and Analysis on Manifolds
Mathematics
Mathematics and Statistics
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Summary:Let X 1 , … , X N be independent random vectors uniformly distributed on an isotropic convex body K ⊂ R n , and let K N be the symmetric convex hull of X i ’s. We show that with high probability L K N ≤ C log ( 2 N / n ) , where C is an absolute constant. This result closes the gap in known estimates in the range C n ≤ N ≤ n 1 + δ . Furthermore, we extend our estimates to the symmetric convex hulls of vectors y 1 X 1 , ⋯ , y N X N , where y = ( y 1 , ⋯ , y N ) is a vector in R N . Finally, we discuss the case of a random vector y .
ISSN:1050-6926
1559-002X
DOI:10.1007/s12220-015-9567-9