Condorcet winners on median spaces

We characterize the outcome of majority voting for single--peaked preferences on median spaces. This large class of preferences covers a variety of multi--dimensional policy spaces including products of lines (e.g.\ grids), trees, and hypercubes. Our main result is the following: If a Condorcet winner (i.e.\ a winner in pairwise majority voting) exists, then it coincides with the appropriately defined median (``the median voter''). This result generalizes previous findings which are either restricted to a one--dimensional policy space or to the assumption that any two voters with the same preference peak must have identical preferences. The result applies to models of spatial competition between two political candidates. A bridge to the graph--theoretic literature is built.