Eliza
O'Reilly
Johns Hopkins University
Operatopes, Operanoids, and Noncommutative zonoids
Abstract.
In this talk, we introduce and study the basic properties of a structured class of convex bodies called operatopes obtained by taking Minkowski sums of affine images of an operator norm ball. This notion generalizes that of zonotopes which are a special class of polytopes obtained by taking Minkowksi sums of line segments. In particular, operatopes include convex bodies that are non-polyhedral. Convex bodies obtained as limits of zonotopes are called zonoids, which can also be viewed as the expectation of a random line segment. Expanding on this interpretation, we can analogously define operanoids as the expectation of a random affine image of an operator norm ball. In studying the properties of operanoids when the dimension of the operator norm ball grows, we also arrive at a natural definition for a class of convex bodies called noncommutative zonoids, and we use the framework of free probability theory to illustrate basic properties and examples. We will discuss a selection of applications of zonoids and the analogous objects of interest for operanoids and noncommutative zonoids as well as open questions. This talk is based on joint work with Venkat Chandrasekaran.