Some convergence properties of Minkowski functionals given by polytopes

In this work we investigate the behavior of the Minkowski Functionals admitted by a sequence of sets which converge to the unit ball ‘from the inside’. We begin in R2 and use this example to build intuition as we extend to the more general Rn case. We prove, in the penultimate chapter, that convergence ‘from the inside’ in this setting is equivalent to two other characterizations of the convergence: a geometric characterization which has to do with the sizes of the faces of each polytope in the sequence converging to zero, and the convergence of the Minkowski functionals defined on the approximating sets to the Euclidean Norm. In the last chapter we explore how we can extend our results to infinite dimensional vector spaces by changing our definition of polytope in that setting, the outlook is bleak. SOME CONVERGENCE PROPERTIES OF MINKOWSKI FUNCTIONALS GIVEN BY POLYTOPES A Thesis Submitted in Partial Fulfillment of the Requirement for the Degree Master of Arts Jesse Moeller University of Northern Iowa May 2016