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

Abstract

We present various results about Euclidean preferences in the plane under ℓ1, ℓ2 and ℓ∞ norms. We show that Euclidean preference profiles under norm ℓ1 are the same as those under norm ℓ∞, and that the maximal size of such profiles for four candidates is 19. Whatever the number of candidates, we prove that at most four distinct candidates can be ranked in last position of an Euclidean preference profile under norm ℓ1, 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 also establish that the maximal size of an Euclidean preference profile under norm ℓ1 is in Θ(m 4 ), i.e., the same order of magnitude as under norm ℓ2. Finally, we provide a new proof that two-dimensional Euclidean preference profiles under norm ℓ2 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 (2011).