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Combinatorial and Discrete Problems in Convex Geometry

Alexander, Matthew R
http://rave.ohiolink.edu/etdc/view?acc_num=kent1508949236617778

Abstract Details

2017, PHD, Kent State University, College of Arts and Sciences / Department of Mathematical Sciences.
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.
Artem Zvavitch (Advisor)
Matthieu Fradelizi (Advisor)
Dmitry Ryabogin (Committee Member)
Fedor Nazarov (Committee Member)
Feodor Dragan (Committee Member)
Jonathan Maletic (Committee Member)
80 p.
Mathematics
convex; geometry; combinatorics; lattices; discrete; slicing inequality; Lipschitz-free space; volume product; Mahlers conjecture; analysis

Recommended Citations

Citations

  • Alexander, M. R. (2017). Combinatorial and Discrete Problems in Convex Geometry [Doctoral dissertation, Kent State University]. OhioLINK Electronic Theses and Dissertations Center. http://rave.ohiolink.edu/etdc/view?acc_num=kent1508949236617778

    APA Style (7th edition)

  • Alexander, Matthew. Combinatorial and Discrete Problems in Convex Geometry. 2017. Kent State University, Doctoral dissertation. OhioLINK Electronic Theses and Dissertations Center, http://rave.ohiolink.edu/etdc/view?acc_num=kent1508949236617778.

    MLA Style (8th edition)

  • Alexander, Matthew. "Combinatorial and Discrete Problems in Convex Geometry." Doctoral dissertation, Kent State University, 2017. http://rave.ohiolink.edu/etdc/view?acc_num=kent1508949236617778

    Chicago Manual of Style (17th edition)

kent1508949236617778
647
© 2017, all rights reserved.
This open access ETD is published by Kent State University and OhioLINK.