When the convexity assumptions fail 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.
If you are only looking for references, try google scholar: https://scholar.google.be/scholar?start=0&amp;q=economics+%22without+convexity%22&amp;hl=nl&amp;as_sdt=0,43.

Among the papers on the first pages of results :


Conley, John P., and Simon Wilkie. "The bargaining problem without convexity: extending the egalitarian and Kalai-Smorodinsky solutions." Economics Letters 36.4 (1991): 365-369.
Yamazaki, Akira. "An equilibrium existence theorem without convexity assumptions." Econometrica: Journal of the Econometric Society (1978): 541-555.
Hurwicz, Leonid, and Stanley Reiter. "On the boundedness of the feasible set without convexity assumptions." International Economic Review (1973): 580-586.
Hamano, Tadashi. "On the existence of equilibria without convexity or free disposal on production technologies." Journal of Mathematical Economics 23.6 (1994): 565-583.
Shapley, Lloyd S., and Martin Shubik. The core of an economy with nonconvex preferences. No. RM-3518-PR. RAND CORP SANTA MONICA CA, 1963.


Then look at the papers who cite the above papers. That usually allows you to uncover other relevant references.