The polytopal norm for a given polytope P and a point a is defined to be `P (a) = min{t : a ∈ tP}, where tP denotes the tth dilation of P. A numerical semigroup is a co-finite subset of the non-negative integers, often denoted in terms of its generators, S = hg1, . . . , gki. For each element n ∈ S, we can associate a set of points, ZS(n), such that for all points a ∈ ZS(n), we have that a · (g1, . . . , gk) = n. Elements of ZS(n) are called factorizations of n ∈ S. In this thesis, we will discuss optimizing polytopal norms defined on sets of factorizations of elements of numerical semigroups. We will present results classifying the eventually quasilinear relationship for max and min polytopal norms for polytopes of dimension k.
