In this paper we characterize strategy-proof voting schemes on Euclide
an spaces. A voting scheme is strategy-proof whenever it is optimal fo
r every agent to report his best alternative. Here the individual pref
erences underlying these best choices are separable and quadratic. It
turns out that a voting scheme is strategy-proof if and only if (alpha
) its range is a closed Cartesian subset of Euclidean space, (beta) th
e outcomes are at a minimal distance to the outcome under a specific c
oordinatewise veto voting scheme, and (gamma) it satisfies some monoto
nicity properties. Neither continuity nor decomposability is implied b
y strategy-proofness, but these are satisfied if we additionally impos
e Pareto-optimality or unanimity.