Oligarchies in spatial environments

In spatial environments we consider social welfare functions satisfying Arrow's requirements, i.e. weak Pareto and independence of irrelevant alternatives. Individual preferences measure distances between alternatives according to the lp-norm (for a fixed 1<=p<=[infinity]). When the policy space is multi-dimensional and the set of alternatives has a non-empty and connected interior and its boundary has no tails, any quasi-transitive welfare function must be oligarchic. As a corollary we obtain that for transitive welfare functions weak Pareto, independence of irrelevant alternatives, and non-dictatorship are inconsistent if the set of alternatives has a non-empty and connected interior and its boundary has no tails.