Euclidean preferences in the plane under $$\varvec{\ell _1},$$ $$\varvec{\ell _2}$$ and $$\varvec{\ell _\infty }$$ norms

Abstract

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 maximum size (in terms of the number of pairwise distinct preferences) of Euclidean preference profiles in \({\mathbb {R}}^2\) 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 the 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 maximum size of a two-dimensional Euclidean preference profile on m candidates under norm \(\ell _1\) is in \(\varTheta (m^4),\) which is the same order of magnitude as the known maximum size 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 inclusion-maximal two-dimensional Euclidean profiles. This proof is a simpler alternative to that proposed by Kamiya et al. (Adv Appl Math 47(2):379–400, 2011).