  We investigate connections between Hardy's inequality in the whole space
$\mathbb{R}^n$ and embedding inequalities for Sobolev-Lorentz spaces. In
particular, we complete previous results due to [A. Alvino, Sulla
diseguaglianza di Sobolev in spazi di Lorentz, (1977)] and [G. Talenti, An
inequality between $u^*$ and $|{\rm{grad}} u^*|$, (1992)] by establishing
optimal embedding inequalities for the Sobolev-Lorentz quasinorm
$\|\nabla\,\cdot\,\|_{p,q}$ also in the range $p < q<\infty$, which remained
essentially open since the work of Alvino. Attainability of the best embedding
constants is also studied, as well as the limiting case when $q=\infty$. Here,
we surprisingly discover that the Hardy inequality is equivalent to the
corresponding Sobolev-Marcinkiewicz embedding inequality. Moreover, the latter
turns out to be attained by the so-called "ghost" extremal functions of
[Brezis-V\'azquez, Blow-up solutions of some nonlinear elliptic problems,
(1977)], in striking contrast with the Hardy inequality, which is never
attained. In this sense, our functional approach seems to be more natural than
the classical Sobolev setting, answering a question raised by Brezis and
V\'azquez.
