In a famous paper from 1974, Hal Varian proved (in Theorem 2.3) that:
He said (after Theorem 2.8) that, if preferences are not convex, then Pareto-efficient envy-free allocations might not exist. As an example, he brings the following economy: there are two agents with the same preferences:
$$u_1(x,y) = u_2(x,y) = \min(x,y)$$
and the total bundle is $(2,1)$.
I do not see how this is a negative example. If we give each agent a bundle of $(1,0.5)$, then the division is envy-free, since the value of every agent is exactly 0.5. It is also Pareto-efficient, since to increase the utility of one agent, it is necessary to give him some y from the other agent, but this hurts the utility of the other agent.
Am I missing something?