Convex Spaces I: Definition and Examples

TL;DR

This paper introduces an abstract, algebraic definition of convex spaces that generalizes classical convex sets, including examples from mathematics and physics, and unifies probabilistic and possibilistic concepts.

Contribution

It provides a new categorical and algebraic framework for convex spaces, extending beyond traditional vector space convexity to include combinatorial structures like semilattices.

Findings

Convex spaces can be characterized as algebras over a finitary Giry monad.

Examples include classical convex sets and semilattices, which are not vector spaces.

Convex spaces unify probabilistic and possibilistic notions of convexity.

Abstract

We propose an abstract definition of convex spaces as sets where one can take convex combinations in a consistent way. A priori, a convex space is an algebra over a finitary version of the Giry monad. We identify the corresponding Lawvere theory as the category from arXiv:0902.2554 and use the results obtained there to extract a concrete definition of convex space in terms of a family of binary operations satisfying certain compatibility conditions. After giving an extensive list of examples of convex sets as they appear throughout mathematics and theoretical physics, we find that there also exist convex spaces that cannot be embedded into a vector space: semilattices are a class of examples of purely combinatorial type. In an information-theoretic interpretation, convex subsets of vector spaces are probabilistic, while semilattices are possibilistic. Convex spaces unify these two concepts.