Small one-dimensional Euclidean preference profiles
研究了少量选民和备选方案下的一维欧几里得偏好分布，发现两个选民的单峰偏好分布总是一维欧几里得的，而最多五个备选方案时，单峰且单交叉的偏好分布才是一维欧几里得的。
Abstract We characterize one-dimensional Euclidean preference profiles with a small number of alternatives and voters. We show that every single-peaked preference profile with two voters is one-dimensional Euclidean, and that every preference profile with up to five alternatives is one-dimensional Euclidean if and only if it is both single-peaked and single-crossing. By the work of Chen et al. (Social Choice and Welfare 48(2):409–432, 2017), we thus obtain that the smallest single-peaked and single-crossing preference profiles that are not one-dimensional Euclidean consist of three voters and six alternatives.