A preference profile with $m$ alternatives and $n$ voters is $d$-Manhattan (resp. $d$-Euclidean) if both the alternatives and the voters can be placed into the $d$-dimensional space such that between each pair of alternatives, every voter prefers the one which has a shorter Manhattan (resp. Euclidean) distance to the voter. Following Bogomolnaia and Laslier [Journal of Mathematical Economics, 2007] and Chen and Grottke [Social Choice and Welfare, 2021] who look at $d$-Euclidean preference profiles, we study which preference profiles are $d$-Manhattan depending on the values $m$ and $n$. First, we show that each preference profile with $m$ alternatives and $n$ voters is $d$-Manhattan whenever $d$ $\geq$ min($n$, $m$-$1$). Second, for $d = 2$, we show that the smallest non $d$-Manhattan preference profile has either three voters and six alternatives, or four voters and five alternatives, or five voters and four alternatives. This is more complex than the case with $d$-Euclidean preferences (see [Bogomolnaia and Laslier, 2007] and [Bulteau and Chen, 2020].

翻译：如果替代品和选民都能在每对替代品之间,每个选民都更喜欢曼哈顿距离较短的替代品。在Bogomolnaia和Lasierer[数学经济学杂志,2007年]之后,以及Chen和Grottke[社会选择和福利杂志,2021年],他们看的是美元-欧元优惠概况,我们研究的是每对替代品之间,每对替代品之间,每个选民都更喜欢与选民距离较短的曼哈顿(Euclidean,2007年美元)的偏好。在Bogomolnaia和Lasier[数学经济学杂志,2007年美元-1美元]之后,以及Chen和Grottke[社会选择和福利杂志,2021年],他们看的是美元-欧元优惠概况,我们研究的偏好范围是美元-曼丹,这取决于价值百万美元和美元。 首先,我们表明,每一份带有1美元替代品和1美元(B)的偏好五个选民或4个选项,2007年和4个选民的偏多。