论文标题
在$ \ ell_1 $，$ \ ell_2 $和$ \ ell_ \ ell_ \ infty $ norms下，飞机上的欧几里得偏好
Euclidean preferences in the plane under $\ell_1$, $\ell_2$ and $\ell_\infty$ norms
论文作者
论文摘要
我们在$ \ ell_1 $，$ \ ell_2 $和$ \ ell _ {\ infty} $ narms下介绍有关飞机中欧几里得首选项的各种结果。当有四个候选人时，我们表明，在规范$ \ ell_1 $或$ \ ell_1 $或$ \ ell _ {\ ell _ {\ infty} $下，飞机上的最大尺寸（根据成对不同的偏好数量）是19。无论是候选人的数量如何$ \ ell_1 $或$ \ ell_ \ infty $，它概括了一维欧几里得偏好的情况（众所周知，最多两个候选人可以最后排名）。我们将此结果推广到$ \ ell_1 $（分别$ \ ell_ \ infty $）的$ 2^d $（分别为$ 2D $），以$ d $ d $ - 二维的欧几里得偏好。我们还确定，根据Norm $ \ ell_1 $在$θ（M^4）$中，二维欧几里得偏好配置文件的最大尺寸，即$ thim $ \ ell_2 $。最后，我们提供了一个新的证据，表明在Norm $ \ ell_2 $下进行的二维欧几里得偏好概况可以用三个选民最大的二维欧几里得个人资料来表征。该证明是Kamiya等人提出的更简单的替代方法。在编码模型的排名模式中，应用数学的进步47（2）：379-400。
We present various results about Euclidean preferences in the plane under $\ell_1$, $\ell_2$ and $\ell_{\infty}$ norms. When there are four candidates, we show that the maximal size (in terms of the number of pairwise distinct preferences) of Euclidean preference profiles in the plane under norm $\ell_1$ or $\ell_{\infty}$ is 19. Whatever the number of candidates, we prove that at most four distinct candidates can be ranked in last position of a two-dimensional Euclidean preference profile under norm $\ell_1$ or $\ell_\infty$, which generalizes the case of one-dimensional Euclidean preferences (for which it is well known that at most two candidates can be ranked last). We generalize this result to $2^d$ (resp. $2d$) for $\ell_1$ (resp. $\ell_\infty$) for $d$-dimensional Euclidean preferences. We also establish that the maximal size of a two-dimensional Euclidean preference profile on $m$ candidates under norm $\ell_1$ is in $Θ(m^4)$, i.e., the same order of magnitude as under norm $\ell_2$. Finally, we provide a new proof that two-dimensional Euclidean preference profiles under norm $\ell_2$ for four candidates can be characterized by three voter-maximal two-dimensional Euclidean profiles. This proof is a simpler alternative to that proposed by Kamiya et al. in Ranking patterns of unfolding models of codimension one, Advances in Applied Mathematics 47(2):379-400.
