preference convexity and existence of equilbria
Consider an economy with two commodities. Production is trivial, $Y=\{0\}$, there is a single consumer with endowment $(1,1)$ whose preferences are represented by the utility function given by $u(x_1,x_2)=\max\{x_1,x_2\}+1/2\cdot x_1 +1/2\cdot x_2$. These preferences are continuous, strictly monotone, but not convex. You can verify that there is no equilibrium in this economy by showing that for no price system, the demand of the consumer equals the supply of $(1,1)$.