We study sets of preferences that are convex with respect to the betweeness relation induced by the Kemeny distance for preferences. It appears that these sets consist of all preferences containing a certain partial ordering and the other way around all preferences containing a given partial ordering form a convex set. Next we consider restricted domains where each agent has a convex set of preferences. Necessary and sufficient conditions are formulated under which a restricted domain admits unanimous, strategy-proof and non-dictatorial choice rules. Loosly speeking it boils down to admitting monotone and non-image-dictatorial decision rules on two alternatives where the other alternatives are completely disregarded.