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Suppose that there is an election with three candidate and an infinite number of voters whose opinion lie in a two-dimensional issue space according to some distribution, and that voter's candidate preferences are based on the euclidean distance of the voter from each candidate.

If the distribution is rotationally symmetrical, it's easy to show that there will be a Condorcet Winner with probability 1 (Basically by showing that two candidates equidistant from the center will split the vote equally in a run-off).

Are there any broader conditions on the opinion distribution in which we can force the probability of the existence of a Condorcet Winner to be 1?

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You could demand that codimension one subspaces have no mass. –  S. Carnahan Sep 25 '11 at 17:27

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