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Volume distribution and the geometry of highdimensional random polytopes

 Author / Creator
 Pivovarov, Peter

This thesis is based on three papers on selected topics in Asymptotic Geometric Analysis. The first paper is about the volume of highdimensional random polytopes; in particular, on polytopes generated by Gaussian random vectors. We consider the question of how many random vertices (or facets) should be sampled in order for such a polytope to capture significant volume. Various criteria for what exactly it means to capture significant volume are discussed. We also study similar problems for random polytopes generated by points on the Euclidean sphere. The second paper is about volume distribution in convex bodies. The first main result is about convex bodies that are (i) symmetric with respect to each of the coordinate hyperplanes and (ii) in isotropic position. We prove that most linear functionals acting on such bodies exhibit superGaussian taildecay. Using known facts about the meanwidth of such bodies, we then deduce strong lower bounds for the volume of certain caps. We also prove a converse statement. Namely, if an arbitrary isotropic convex body (not necessarily satisfying the symmetry assumption (i)) exhibits similar capbehavior, then one can bound its meanwidth. The third paper is about random polytopes generated by sampling points according to multiple logconcave probability measures. We prove related estimates for random determinants and give applications to several geometric inequalities; these include estimates on the volumeradius of random zonotopes and Hadamard's inequality for random matrices.

 Subjects / Keywords

 Graduation date
 201011

 Type of Item
 Thesis

 Degree
 Doctor of Philosophy

 License
 This thesis is made available by the University of Alberta Libraries with permission of the copyright owner solely for noncommercial purposes. This thesis, or any portion thereof, may not otherwise be copied or reproduced without the written consent of the copyright owner, except to the extent permitted by Canadian copyright law.