A preference profile (i.e., a collection of linear preference orders of the voters over a set of alternatives) with $m$ alternatives and $n$ voters is $d$-Manhattan (resp. $d$-Euclidean) if both the alternatives and the voters can be placed into a $d$-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. We study how $d$-Manhattan preference profiles depend on the values $m$ and $n$. First, we provide explicit constructions to show that each preference profile with $m$ alternatives and $n$ voters is $d$-Manhattan whenever $d \ge \min(n, m - 1)$. We further extend this positive result for other $p$-norms with $p \in R_{\ge 1} \cup \{\infty\}$. Second, for $d = 2$, we develop forbidden substructures-preference patterns among small sets of voters that constrain any 2-Manhattan embedding -- and use them to show that the smallest non-2-Manhattan preference profile has either 3 voters and 6 alternatives, or 4 voters and 5 alternatives, or 5 voters and 4 alternatives. This is more complex than the case with $d$-Euclidean preferences (see (Bogomolnaia and Laslier, 2007) and (Bulteau and Chen, 2022)). We also show that $d$-Manhattan preferences imply $(2d-1)$-dimensional single-peakedness, while 2-Manhattanness is incomparable with single-peakedness and single-crossingness.

翻译：偏好画像（即选民对一组备选方案的线性偏好序的集合）包含$m$个备选方案和$n$个选民，称为$d$-曼哈顿偏好（或$d$-欧几里得偏好），当且仅当备选方案与选民均可置于$d$维空间中，使得每对备选方案之间，每位选民偏好与其曼哈顿距离（或欧几里得距离）更短的备选方案。我们研究$d$-曼哈顿偏好如何依赖于参数$m$和$n$。首先，我们给出显式构造，证明当$d \ge \min(n, m-1)$时，任何包含$m$个备选方案和$n$个选民的偏好画像均为$d$-曼哈顿偏好。我们进一步将这一正向结论推广至其他$p$-范数（其中$p \in R_{\ge 1} \cup \{\infty\}$）。其次，针对$d=2$，我们发展了禁止子结构——即约束任意2-曼哈顿嵌入的小规模选民集合的偏好模式——并利用它们证明：最小的非2-曼哈顿偏好画像包含3名选民与6个备选方案，或4名选民与5个备选方案，或5名选民与4个备选方案。这一结果比$d$-欧几里得偏好的情况更为复杂（参见(Bogomolnaia and Laslier, 2007)与(Bulteau and Chen, 2022)）。我们还证明，$d$-曼哈顿偏好蕴含$(2d-1)$-维单峰性，而2-曼哈顿性与单峰性及单交叉性不可比较。