A voting scheme assigns to each profile of alternatives reported by n individuals a compromise alternative. A voting scheme is strategy-proof if no individual is better off by lying, i.e., not reporting a best alternative. In this paper the main results concern the case where the set of alternatives is the Euclidean plane and the preferences are Euclidean. It is shown that for strategy-proof voting schemes continuity is equivalent to convexity of the range of the voting scheme. Using a result by Kim and Roush, this leads to characterizations of surjective or unanimous, anonymous, strategy-proof voting schemes. Furthermore, the paper contains an extensive discussion of related results in the area.
