In a model with finitely many agents who have single-dipped Euclidean preferences on a polytope in the Euclidean plane, a rule assigns to each profile of reported dips a point of the polytope. A single-best point is a point which is the unique point at maximal distance from some other point of the polytope. It is proved that any strategy-proof and Pareto optimal rule is a dictatorship unless the polytope has exactly two single- best points or it has exactly four single-best points which form the vertices of a rectangle. In the latter cases strategy-proof and Pareto optimal rules can be obtained by committee voting (simple games) between the single- best alternatives. This framework models situations where public bads such as garbage dumping grounds or nuclear plants have to be located within a confined region.