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2-Dimensional Euclidean Preferences

A preference profile with m alternatives and n voters is 2-dimensional Euclidean if both the alternatives and the voters can be placed into a 2-dimensional space such that for each pair of alternatives, every voter prefers the one which has a shorter Euclidean distance to the voter. We study how 2-dimensional Euclidean preference profiles depend on the values m and n. We find that any profile with at most two voters or at most three alternatives is 2-dimensional Euclidean while for three voters, we can show this property for up to seven alternatives. The results are tight in terms of Bogomolnaia and Laslier [2, Proposition 15(1)].