Convex bodies generated by sublinear expectations of random vectors

Abstract

We show that many well-known transforms in convex geometry (in particular, centroid body, convex floating body, and Ulam floating body) are special instances of a general construction, relying on applying sublinear expectations to random vectors in Euclidean space. We identify the dual representation of such convex bodies and describe a construction that serves as a building block for all so defined convex bodies. Sublinear expectations are studied in mathematical finance within the theory of risk measures. In this way, tools from mathematical finance yield a whole variety of new geometric constructions.