### Convex Spaces

#### Posted by David Corfield

It’s good to see other people talking about things we chat about here. So I was interested to see today Tobias Fritz’s paper Convex Spaces I: Definition and Examples:

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.

This deals with issues raised in Tom’s thread (referenced by Fritz) and in this recent thread.

Fritz goes on to discuss examples of convexity, including KMS states and torus actions on symplectic manifolds.

## Re: Convex Spaces

Thanks for the publicity!

I sure would like to know what you guys think about this abstract approach. In particular, how natural are the convex spaces of combinatorial type? Somehow it still seems slightly contrived to me due to the discontinuities involved. On the other hand, I believe that they could unify concepts previously thought to be independent: e.g. today I realized that this seems to be the case for Fourier-Motzkin elimination and the resolution technique in Boolean logic. Both of these work by eliminating variables one at a time.