Single-peaked preferences, endowments and population-monotonicity

We consider economies with a finite set of agents with single-peaked preferences and with private endowments of an infinitely divisible commodity. We look for solutions that distribute fairly across agents the gains made possible by redistributions of the commodity under the consideration of changes in the number of agents. We show that the uniform rule is the only selection from the weak no-envy and Pareto efficient solution satisfying one-sided population-monotonicity.