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. / Mathematics
| Identifer | oai:union.ndltd.org:LACETR/oai:collectionscanada.gc.ca:AEU.10048/1170 |
| Date | 11 1900 |
| Creators | Pivovarov, Peter |
| Contributors | Tomczak-Jaegermann, Nicole (Mathematical and Statistical Sciences), Litvak, Alexander (Mathematical and Statistical Sciences), Troitsky, Vladimir (Mathematical and Statistical Sciences), Lau, Anthony To-Ming (Mathematical and Statistical Sciences), Yaskin, Vlad (Mathematical and Statistical Sciences), Stewart, Lorna (Computing Sciences), Werner, Elisabeth (Mathematics, Case Western Reserve University) |
| Source Sets | Library and Archives Canada ETDs Repository / Centre d'archives des thèses électroniques de Bibliothèque et Archives Canada |
| Language | English |
| Detected Language | English |
| Type | Thesis |
| Format | 825680 bytes, application/pdf |
| Relation | P. Pivovarov. (2007). Studia Math. 183, 15-34., P. Pivovarov. Math. Proc. Cambridge Philos. Soc., P. Pivovarov. Geom. Dedicata. |
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