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.

中文翻译：

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小的一维欧几里得偏好配置文件

我们用少量备选方案和选民来表征一维欧几里得偏好配置文件。我们表明，具有两个选民的每个单峰偏好配置文件都是一维欧几里得，并且每个具有多达五个选项的偏好配置文件都是一维欧几里得，当且仅当它既是单峰又是单交叉。通过 Chen 等人的工作。(Social Choice and Welfare 48(2):409–432, 2017)，因此我们获得了非一维欧几里得的最小单峰和单交叉偏好配置文件由三个选民和六个备选方案组成。