  We present a formula for the Fourier transforms of order statistics in $\Bbb
R^n$ showing that all these Fourier transforms are equal up to a constant
multiple outside the coordinate planes in $\Bbb R^n.$
  For $a_1\geq ... \geq a_n\ge0$ and $q>0,$ denote by $\ell_{w,q}^n$ the
$n$-dimensional Lorentz space with the norm $\|(x_1,...,x_n)\| = (a_1
(x_1^{*})^q +...+ a_n (x_n^{*})^q)^{1/q}$, where $(x_1^{*},...,x_n^{*})$ is the
non-increasing permutation of the numbers $|x_1|,...,|x_n|.$ We use the above
mentioned formula and the Fourier transform criterion of isometric
embeddability of Banach spaces into $L_q$ \cite{10} to prove that, for $n\geq
3$ and $q\leq 1,$ the space $\ell_{w,q}^n$ is isometric to a subspace of $L_q$
if and only if the numbers $a_1,...,a_n$ form an arithmetic progression. For
$q>1,$ all the numbers $a_i$ must be equal so that $\ell_{w,q}^n = \ell_q^n.$
Consequently, the Lorentz function space $L_{w,q}(0,1)$ is isometric to a
subspace of $L_q$ if and only if {\it either} $0<q<\infty$ and the weight $w$
is a constant function (so that $L_{w,q}= L_q$), {\it or} $q\le 1$ and $w(t)$
is a decreasing linear function.
  Finally, we relate our results to the theory of positive definite functions.
