Files in this item



application/pdfMilos_Curcic.pdf (2MB)
(no description provided)PDF


Title:Lattice polytopes with distinct pair-sums
Author(s):Curcic, Milos
Director of Research:Reznick, Bruce A.
Doctoral Committee Chair(s):Yong, Alexander
Doctoral Committee Member(s):Reznick, Bruce A.; Athreya, Jayadev S.; Carpenter, Bruce
Department / Program:Mathematics
Degree Granting Institution:University of Illinois at Urbana-Champaign
Subject(s):lattice polytopes
distinct pair-sums
distinct pair-sums volume
clean tetrahedron
Abstract:Let P be a lattice polytope in R^d, the convex hull of a finite set in Z^d, and let L(P) = P intersection Z^d = {v1,..., vN}, where N = |L(P)|. We call P a distinct pair-sum or dps polytope if L(P)+L(P) contains N + N choose 2 distinct points. A maximal dps polytope in R^d is a dps polytope for which N = 2^d. In 2002, Choi, Lam and Reznick presented a method for constructing maximal dps polytopes in R^d for every d. They showed that the maximal dps polygons in R^2 are equivalent under unimodular maps to the triangle with vertices (0, 1), (1, 2), (2, 0), and presented two examples of inequivalent maximal dps polyhedra in R^3. The main focus of this dissertation is to examine the combinatorial and geometric structure of dps polyhedra in R^3 and to classify them up to unimodular transformations. Using a computer search and a theorem of Pikhurko, we exploit the relationship between the maximal dps polygons and maximal dps polyhedra to find all maximal dps tetrahedra up to unimodular transformations. We also present partial results on maximal dps polyhedra with more than four vertices.
Issue Date:2013-02-03
Rights Information:Copyright 2012 Milos Curcic
Date Available in IDEALS:2013-02-03
Date Deposited:2012-12

This item appears in the following Collection(s)

Item Statistics