PLURIPOTENTIAL THEORY ASSOCIATED WITH CONVEX BODIES AND RELATED NUMERICSPLURIPOTENTIAL THEORY ASSOCIATED WITH CONVEX BODIES AND RELATED NUMERICS

Abstract

We explore aspects of pluripotential theory generalized to the convex body associated case. We focus on theoretical justifications of numerical approximations to important objects and quantities. We implement some of these constructions using the Python programming language. Pluripotential theory is a branch of the study of several complex variables expanding univariate potential theory. Many of the foundational proofs rely on the precise lexicographical ordering of the monomials and the standard notion of degree. However, it has recently been shown that different orderings have value in polynomial approximation [Tre17]. This dissertation takes part in the effort to generalize pluripotential theory to the case of these novel orderings and new notions of degree, which are associated with a given convex body. We begin with a motivational section on why pluripotential theory associated with a convex body is a valuable course of study and provide the necessary background in pluripotential theory, convex body associated polynomials, and associated orderings. We prove a theorem dividing convex bodies into classes based on whether or not an “additive and nested” ordering can be constructed. Further, we showcase a counterexample with a regularization issue in the convex body case. We then discuss finite point sequences associated with a given compact set. After providing the classic definitions, we generalize numerical algorithms to generate points associated with the polynomials given by a convex body as well as showcasing a numerical implementation of these algorithms in Python 3 [VRD09]. We provide a chapter generalizing numerical approximation algorithms to the convex body associated case: these algorithms allow us to approximate several important pluripotential theoretic constructions. We discuss a related problem from optimal design theory: proving the global convergence of a measure-theoretic Silvey–Titterington–Torsney algorithm. We have implemented the algorithm in Python for some cases, and much progress has been made on the proof of convergence. Lastly, an appendix chapter is included showing that the measure-theoretic framing of the optimal design problem is valid. A second appendix chapter provides the full implementation of the algorithms to construct finite point sequences and arrays.