Barberá, Massó and Serizawa (1998) provided full characterization for class of strategy-proof social choice functions for societies where the set of alternatives is any full dimensional compact subset of a Euclidean space and all voters have generalized single-peaked preferences. They proved that this class is composed by generalized median voter schemes satisfying an additional condition, called the “intersection property”. But according to their results in order to understand whether any generalized median voter scheme satisfies intersection property for given set of alternatives or not it was necessary to check all the alternatives from the set of unfeasible alternatives - addition of the set of feasible alternatives to minimal Cartesian product range, containing this set. So the number of alternatives to be checked, was infinite. In this paper it is proved, that it is enough to check finite number of alternatives from the set of unfeasible alternatives and constructive algorithm to determine alternatives that should be checked is provided.