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Some Inequalities in Convex Geometry Open Access


Other title
transversal numbers
convex geometry
measures of symmetry
Banach-Mazur distance
independence numbers
Type of item
Degree grantor
University of Alberta
Author or creator
Taschuk, Steven J.
Supervisor and department
Litvak, Alexander (Mathematical and Statistical Sciences)
Tomczak-Jaegermann, Nicole (Mathematical and Statistical Sciences)
Examining committee member and department
Litvak, Alexander (Mathematical and Statistical Sciences)
Troitsky, Vladimir (Mathematical and Statistical Sciences)
Hillen, Thomas (Mathematical and Statistical Sciences)
Tomczak-Jaegermann, Nicole (Mathematical and Statistical Sciences)
Zvavitch, Artem (Mathematics and Computer Sciences, Kent State University)
Department of Mathematical and Statistical Sciences
Date accepted
Graduation date
Doctor of Philosophy
Degree level
We present some inequalities in convex geometry falling under the broad theme of quantifying complexity, or deviation from particularly pleasant geometric conditions: we give an upper bound for the Banach--Mazur distance between an origin-symmetric convex body and the $n$-dimensional cube which improves known bounds when n is at least 3 and is "small"; we give the best known upper and lower bounds (for high dimensions) for the maximum number of points needed to hit every member of an intersecting family of positive homothets (or translates) of a convex body, a number which quantifies the complexity of the family's intersections; we give an exact upper bound on the VC-dimension (a measure of combinatorial complexity) of families of positive homothets (or translates) of a convex body in the plane, and show that no such upper bound exists in any higher dimension; finally, we introduce a novel volumetric functional on convex bodies which quantifies deviation from central symmetry, establish the fundamental properties of this functional, and relate it to classical volumetric measures of symmetry.
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 these terms. The author reserves all other publication and other rights in association with the copyright in the thesis and, except as herein before 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.
Citation for previous publication
Steven Taschuk. The Banach-Mazur distance to the cube in low dimensions. Discrete Comput. Geom., 46:175-183, 2011.Marton Naszodi and Steven Taschuk. On the transversal number and VC-dimension of families of positive homothets of a convex body. Discrete Math., 310:77-82, 2010.

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