Some problems from convex geometry and geometric tomography

• Author / Creator

  Stephen, Matthew A.

• We address several related problems from convex geometry and geometric tomography, which separate along two main themes. The content of this thesis is based on five papers, either published or submitted.

  The first theme concerns the origin-symmetry and unique determination of convex bodies. A convex body K is a compact and convex subset in n-dimensional Euclidean space with non-empty interior. We say K is origin-symmetric if it is equal to its reflection through the origin, that is K=-K. Makai, Martini, and Odor have shown that a convex body is necessarily origin-symmetric if every hyperplane section through the origin has maximal (n-1)-dimensional volume amongst all parallel sections. We prove a stability version of their result.

  Recently, Meyer and Reisner associated with every convex body K a new set, which they call the convex intersection body of K. It follows from previously known results that two origin-symmetric convex bodies coincide whenever their convex intersection bodies coincide. Removing the assumption of origin-symmetry, we show that Meyer and Reisner's convex intersection body does not uniquely determine a convex body up to congruency.

  A convex polytope P is a convex body which is the convex hull of finitely many points. We show that P must be origin-symmetric if every hyperplane section through the origin has maximal (n-2)-dimensional surface area amongst all parallel sections. This gives partial confirmation to a conjecture made by Makai, Martini, and Odor.

  Our second theme concerns extensions of Grunbaum's inequality, which gives a sharp lower bound for the volume of each half of a convex body that is split by a hyperplane through its centroid. In particular, we generalize this inequality to the orthogonal projections of a convex body onto subspaces, and the intersections of a convex body with subspaces through its centroid.

• Subjects / Keywords

  • volume inequality
  • sections
  • convex body
  • unique determination
  • origin-symmetry
  • geometric tomography
  • polytope
  • convex geometry
  • centroid
  • projections

• Graduation date

  Fall 2018

• Type of Item

  Thesis

• Degree

  Doctor of Philosophy

• DOI

  https://doi.org/10.7939/R3NC5SV3M

• License

  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.