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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

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 (6th 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 (16th 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 (7th 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

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