Hotelling's Law and the Median Voter Theorem break down in higher dimensional space, just like everything else does.
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The first time I got some intuition for that was in Bendor et al’s A Behavioral Theory of Elections. They had this Voronoi tessellation simulation that suddenly made everything seem much more daunting to me.
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I don't know enough math to really understand some of the finer mathematical critiques, but I remember in his book itself he's like 'well, if it turns out you DON'T have a normal distribution...none of this works.'
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