Euclidean preferences in the plane under l1, l2 and l∞ norms

We present various results about Euclidean preferences in the plane under l(1), l(2) and l(infinity) 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 R-2 under norm l(1) or l(infinity) 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 l(1) or l(infinity), 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 l(1) (resp. l(infinity)) for d-dimensional Euclidean preferences. We also establish that the maximum size of a two-dimensional Euclidean preference profile on m candidates under norm l(1) is in Theta(m(4)), which is the same order of magnitude as the known maximum size under norm l(2). Finally, we provide a new proof that two-dimensional Euclidean preference profiles under norm l(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).