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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 distinctpreferences) of Euclidean preference profiles in the plane under norm $\ell_1$or $\ell_{\infty}$ is 19. Whatever the number of candidates, we prove that atmost four distinct candidates can be ranked in last position of atwo-dimensional Euclidean preference profile under norm $\ell_1$ or$\ell_\infty$, which generalizes the case of one-dimensional Euclideanpreferences (for which it is well known that at most two candidates can beranked last). We generalize this result to $2^d$ (resp. $2d$) for $\ell_1$(resp. $\ell_\infty$) for $d$-dimensional Euclidean preferences. We alsoestablish that the maximal size of a two-dimensional Euclidean preferenceprofile on $m$ candidates under norm $\ell_1$ is in $\Theta(m^4)$, i.e., thesame order of magnitude as under norm $\ell_2$. Finally, we provide a new proofthat two-dimensional Euclidean preference profiles under norm $\ell_2$ for fourcandidates can be characterized by three voter-maximal two-dimensionalEuclidean profiles. This proof is a simpler alternative to that proposed byKamiya et al. in Ranking patterns of unfolding models of codimension one,Advances in Applied Mathematics 47(2):379-400.

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