Formal Adventures in Convex and Conical Spaces

Convex sets appear in various mathematical theories, and are used to define notions such as convex functions and hulls. As an abstraction from the usual definition of convex sets in vector spaces, we formalize in Coq an intrinsic axiomatization of convex sets, namely convex spaces, based on an operation taking barycenters of points. A convex space corresponds to a specific type that does not refer to a surrounding vector space. This simplifies the definitions of functions on it. We show applications including the convexity of information-theoretic functions defined over types of distributions. We also show how convex spaces are embedded in conical spaces, which are abstract real cones, and use the embedding as an effective device to ease calculations.