  The Wigner function $W_N({\bf x}, {\bf p})$ is a useful quantity to
characterize the quantum fluctuations of an $N$-body system in its phase space.
Here we study $W_N({\bf x}, {\bf p})$ for $N$ noninteracting spinless fermions
in a $d$-dimensional spherical hard box of radius $R$ at temperature $T=0$. In
the large $N$ limit, the local density approximation (LDA) predicts that
$W_N({\bf x}, {\bf p}) \approx 1/(2 \pi \hbar)^d$ inside a finite region of the
$({\bf x}, {\bf p})$ plane, namely for $|{\bf x}| < R$ and $|{\bf p}| < k_F$
where $k_F$ is the Fermi momentum, while $W_N({\bf x}, {\bf p})$ vanishes
outside this region, or "droplet", on a scale determined by quantum
fluctuations. In this paper we investigate systematically, in this quantum
region, the structure of the Wigner function along the edge of this droplet,
called the Fermi surf. In one dimension, we find that there are three distinct
edge regions along the Fermi surf and we compute exactly the associated
nontrivial scaling functions in each regime. We also study the momentum
distribution $\hat \rho_N(p)$ and find a striking algebraic tail for very large
momenta $\hat \rho_N(p) \propto 1/p^4$, well beyond $k_F$, reminiscent of a
similar tail found in interacting quantum systems (discussed in the context of
Tan's relation). We then generalize these results to higher $d$ and find,
remarkably, that the scaling function close to the edge of the box is
universal, i.e., independent of the dimension~$d$.
