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Some Inequalities in Convex Geometry

  • Author / Creator
    Taschuk, Steven J.
  • 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.

  • Subjects / Keywords
  • Graduation date
    2013-11
  • Type of Item
    Thesis
  • Degree
    Doctor of Philosophy
  • DOI
    https://doi.org/10.7939/R3FH6B
  • 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.
  • Language
    English
  • Institution
    University of Alberta
  • Degree level
    Doctoral
  • Department
    • Department of Mathematical and Statistical Sciences
  • Specialization
    • Mathematics
  • Supervisor / co-supervisor and their department(s)
    • Tomczak-Jaegermann, Nicole (Mathematical and Statistical Sciences)
    • Litvak, Alexander (Mathematical and Statistical Sciences)
  • Examining committee members and their departments
    • Tomczak-Jaegermann, Nicole (Mathematical and Statistical Sciences)
    • Zvavitch, Artem (Mathematics and Computer Sciences, Kent State University)
    • Troitsky, Vladimir (Mathematical and Statistical Sciences)
    • Litvak, Alexander (Mathematical and Statistical Sciences)
    • Hillen, Thomas (Mathematical and Statistical Sciences)