In this paper we characterize strategy-proof voting schemes on Euclidean spaces. A voting scheme is strategy-proof whenever it is optimal for every agent to report his best alternative. Here the individual preferences underlying these best choices are separable and quadratic. It turns out that a voting scheme is strategy-proof if and only if (a) its range is a closed Cartesian subset of Euclidean space, (ß) the outcomes are at a minimal distance to the outcome under a specific coordinatewise veto voting scheme, and (¿) it satisfies some monotonicity properties. Neither continuity nor decomposability is implied by strategy-proofness, but these are satisfied if we additionally impose Pareto-optimality or unanimity.