When the convexity assumptions fail?

Question

We constantly assume convexity in economics. Convexity of production sets, convexity of preferences etc. Is there literature that addresses when convexities fail? For example, is there any literature that addresses the existence of equilibria when agents have non-convex preferences? What are the problems associated with non-convex production sets? It just seems like GE theorem relies heavily on convexity assumptions and I have yet to hear anyone reasonably addresses how to approach situations in which these assumptions fail. I would appreciate any good references or any general discussion of non-convexity from both a consumer and a producer perspective.

Answer 1