Euclidean preferences in the plane under $\ell_1$, $\ell_2$ and $\ell_\infty$ norms

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 $\Theta(m^4)$, 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.