I've been reading a few papers on generalized probabilistic theories, and have been struggling through proofs of some results that involve use of convexity and group theory, e.g. this paper on bit symmetry and many others. Is there a set of introductory lecture notes on convexity (as used in quantum theory) online that anyone can refer me to?
I am currently dealing with generalized probabilistic theories too, and I had the same problem. For example, I read papers like this one: http://arxiv.org/abs/1012.1215 I don't know a good online reference for this kind of math, but a book that I liked reading very much was
This book helped me in getting used to the cone structure of generalized probabilistic theories. It tells you, for example, what an extreme point is (which corresponds to pure states), what a cone base is (which corresponds to the normalized states), what an order unit is (which corresponds to the unit effect), what an order interval of the dual cone is (which corresponds to the set of effects) and so on. I think reading the first chapter, part of the second chapter and part of the third chapter of this book might be helpful for you. The other chapters are probably too mathematical to be helpful in this context.