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Volume distribution and the geometry of high-dimensional random polytopes Open Access


Other title
isotropic constants
log-concave measures
random polytopes
geometric inequalities
convex bodies
Type of item
Degree grantor
University of Alberta
Author or creator
Pivovarov, Peter
Supervisor and department
Tomczak-Jaegermann, Nicole (Mathematical and Statistical Sciences)
Examining committee member and department
Litvak, Alexander (Mathematical and Statistical Sciences)
Stewart, Lorna (Computing Sciences)
Werner, Elisabeth (Mathematics, Case Western Reserve University)
Troitsky, Vladimir (Mathematical and Statistical Sciences)
Lau, Anthony To-Ming (Mathematical and Statistical Sciences)
Yaskin, Vlad (Mathematical and Statistical Sciences)
Department of Mathematical and Statistical Sciences

Date accepted
Graduation date
Doctor of Philosophy
Degree level
This thesis is based on three papers on selected topics in Asymptotic Geometric Analysis. The first paper is about the volume of high-dimensional 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 super-Gaussian tail-decay. Using known facts about the mean-width 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 cap-behavior, then one can bound its mean-width. The third paper is about random polytopes generated by sampling points according to multiple log-concave probability measures. We prove related estimates for random determinants and give applications to several geometric inequalities; these include estimates on the volume-radius of random zonotopes and Hadamard's inequality for random matrices.
License granted by Peter Pivovarov ( on 2010-05-21T21:35:35Z (GMT): Permission is hereby granted to the University of Alberta Libraries to reproduce single copies of this thesis and to lend or sell such copies for private, scholarly or scientific research purposes only. Where the thesis is converted to, or otherwise made available in digital form, the University of Alberta will advise potential users of the thesis of the above terms. The author reserves all other publication and other rights in association with the copyright in the thesis, and except as herein provided, neither the thesis nor any substantial portion thereof may be printed or otherwise reproduced in any material form whatsoever without the author's prior written permission.
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File title: Introduction
File title: High-dimensional random polytopes, Ph. D. Thesis
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