Question: In an exchange economy with $2$ goods and $m$ identical Households where each household has utility function $u(x\_1, x\_2)$, together with positive endowments. If preferences are not convex, then why a general equilibrium may fail to exist?

I am thinking about this diagram: ![enter image description here](https://i.stack.imgur.com/bLgru.png)

Answer: 
Consider the following Edgeworth box. Both Ann and Bob have non convex preferences. The set of allocations prefered by Bob to the initial endowment is colored gray, the set of allocations preferred by Ann to the initial endowment is colored blue. The initial endowment is actually Pareto efficient. Since, given any prices, both could consume their endowment, and making someone better off would entail making the other person worse off, in every Equilibrium, both Ann and Bob would simply consume their endowment. Any equilibrium price system would be represented by a line that separates the gray and the blue area, which is obviously not possible here.

![enter image description here](https://i.stack.imgur.com/uvwnR.png)

Of course, if preferences are convex and the initial endowment is Pareto efficient (and a number of technical assumptions, the strongest one being monotonicity) are satisfied, we could separate these sets and thereby get an equilibrium. This is essentially what the proof of the second welfare theorem does.
Convexity is assumed to make the theory more interesting. Try drawing an Edgeworth box for the case of two households. If preferences are convex, that means there are allocations that are Pareto-superior to the initial endowment. The MRS of the two households are such that they are both willing to trade away from their endowments---household 1 values good $x\_1$ while household 2 values good $x\_2$, hence trade happens. (The resulting competitive equilibria lie on the contract curve.)

If preferences are not convex, there may not be trade, which makes the initial endowment the sole competitive equilibrium.