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
  • Graduation date
    Fall 2018
  • Type of Item
  • Degree
    Doctor of Philosophy
  • DOI
  • License
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