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)].

翻译：如果可以将备选方案和选民放在一个二维空间,那么,每个选民都更喜欢与选民距离较短的备选方案。我们研究了二维欧几里得偏爱模式如何取决于数值m和n。我们发现,最多有两个选民或最多三个备选方案的备选方案都属于二维欧几里德,而对于三个选民,我们可以展示出最多七个备选方案的属性。结果与Bogoomolnaia和Lasier相近[2, Proposition 15(1)]。