  The assumption that voters' preferences share some common structure is a
standard way to circumvent NP-hardness results in social choice problems. While
the Kemeny ranking problem is NP-hard in the general case, it is known to
become easy if the preferences are 1-dimensional Euclidean. In this note, we
prove that the Kemeny ranking problem remains NP-hard for $k$-dimensional
Euclidean preferences with $k\!\ge\!2$ under norms $\ell_1$, $\ell_2$ and
$\ell_\infty$, by showing that any weighted tournament (resp. weighted
bipartite tournament) with weights of same parity (resp. even weights) is
inducible as the weighted majority tournament of a profile of 2-Euclidean
preferences under norm $\ell_2$ (resp. $\ell_1,\ell_{\infty}$), computable in
polynomial time. More generally, this result regarding weighted tournaments
implies, essentially, that hardness results relying on the (weighted) majority
tournament that hold in the general case (e.g., NP-hardness of Slater ranking)
are still true for 2-dimensional Euclidean preferences.
