Euclidean preferences in the plane under $$\varvec{\ell _1},$$ ℓ 1 , $$\varvec{\ell _2}$$ ℓ 2 and $$\varvec{\ell _\infty }$$ ℓ ∞ norms

We present various results about Euclidean preferences in the plane under $$\ell _1,$$ ℓ 1 , $$\ell _2$$ ℓ 2 and $$\ell _{\infty }$$ ℓ ∞ 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 $${\mathbb {R}}^2$$ R 2 under norm $$\ell _1$$ ℓ 1 or $$\ell _{\infty }$$ ℓ ∞ 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 $$\ell _1$$ ℓ 1 or $$\ell _\infty ,$$ ℓ ∞ , 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 $$2^d$$ 2 d (resp. 2d) for $$\ell _1$$ ℓ 1 (resp. $$\ell _\infty $$ ℓ ∞ ) for d-dimensional Euclidean preferences. We also establish that the maximum size of a two-dimensional Euclidean preference profile on m candidates under norm $$\ell _1$$ ℓ 1 is in $$\varTheta (m^4),$$ Θ ( m 4 ) , which is the same order of magnitude as the known maximum size under norm $$\ell _2.$$ ℓ 2 . Finally, we provide a new proof that two-dimensional Euclidean preference profiles under norm $$\ell _2$$ 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).