Volume distribution and the geometry of high-dimensional 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 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.

• Subjects / Keywords

  • Isotropic constants
  • Random polytopes
  • Log-concave measures
  • Geometric inequalities
  • Convex bodies

• Graduation date

  Fall 2010

• Type of Item

  Thesis

• Degree

  Doctor of Philosophy

• DOI

  https://doi.org/10.7939/R36C8J

• License

  This thesis is made available by the University of Alberta Libraries with permission of the copyright owner solely for non-commercial 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.