  The classical Losev-Manin space can be interpreted as a toric
compactification of the moduli space of $n$ points in the affine line modulo
translation and scaling. Motivated by this, we study its higher-dimensional
toric counterparts, which compactify the moduli space of $n$ distinct labeled
points in affine space modulo translation and scaling. We show that these
moduli spaces are a fibration over a product of projective spaces -- with
fibers isomorphic to the Losev-Manin space -- and that they are isomorphic to
the normalization of a Chow quotient. Moreover, we present a criterion to
decide whether the blow-up of a toric variety along the closure of a subtorus
is a Mori dream space. As an application, we demonstrate that a related
generalization of the moduli space of pointed rational curves proposed by Chen,
Gibney, and Krashen is not a Mori dream space when the number of points is at
least nine, regardless of the dimension.
