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Kent State University

1. Alexander, Matthew R. Combinatorial and Discrete Problems in Convex Geometry.

Degree: PhD, College of Arts and Sciences / Department of Mathematical Science, 2017, Kent State University

URL: http://rave.ohiolink.edu/etdc/view?acc_num=kent1508949236617778

In this dissertation we study discrete versions of
several classical problems in convex geometry. First among these is
a natural extension of Alexander Koldobsky's slicing inequality,
which is an equivalent question to the isomorphic version of the
Busemann-Petty problem for arbitrary measures. For our study we
take the discrete measure of the cardinality of the lattice points
inside a body. Our results give an asymptotic bound depending only
on the dimension, and that the bound must be such in the case of
unconditional bodies. We also investigate questions related to the
volume product of convex bodies. In particular, we explore what the
maximal volume product is for polytopes with a fixed number of
vertices. It turns out that the body which yields the maximal
volume product must be a simplex. Finally, we explore a more
discrete version of the volume product that comes from associating
the space of Lipschitz functions over a metric space to a symmetric
polytope with conditions on its vertices, called the unit ball of
the Lipschitz-free space. We then relate the maximal and minimal
balls of such spaces to special graphs associated to the metric
space.
*Advisors/Committee Members: Zvavitch, Artem (Advisor), Fradelizi, Matthieu (Advisor).*

Subjects/Keywords: Mathematics; convex; geometry; combinatorics; lattices; discrete; slicing inequality; Lipschitz-free space; volume product; Mahlers conjecture; analysis

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APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6^{th} Edition):

Alexander, M. R. (2017). Combinatorial and Discrete Problems in Convex Geometry. (Doctoral Dissertation). Kent State University. Retrieved from http://rave.ohiolink.edu/etdc/view?acc_num=kent1508949236617778

Chicago Manual of Style (16^{th} Edition):

Alexander, Matthew R. “Combinatorial and Discrete Problems in Convex Geometry.” 2017. Doctoral Dissertation, Kent State University. Accessed November 17, 2017. http://rave.ohiolink.edu/etdc/view?acc_num=kent1508949236617778.

MLA Handbook (7^{th} Edition):

Alexander, Matthew R. “Combinatorial and Discrete Problems in Convex Geometry.” 2017. Web. 17 Nov 2017.

Vancouver:

Alexander MR. Combinatorial and Discrete Problems in Convex Geometry. [Internet] [Doctoral dissertation]. Kent State University; 2017. [cited 2017 Nov 17]. Available from: http://rave.ohiolink.edu/etdc/view?acc_num=kent1508949236617778.

Council of Science Editors:

Alexander MR. Combinatorial and Discrete Problems in Convex Geometry. [Doctoral Dissertation]. Kent State University; 2017. Available from: http://rave.ohiolink.edu/etdc/view?acc_num=kent1508949236617778