  Let $K$ be a convex polyhedron and $\mathscr F$ its Wulff energy, and let
$\mathscr C(K)$ denote the set of convex polyhedra close to $K$ whose faces are
parallel to those of $K$. We show that, for sufficiently small $\epsilon$, all
$\epsilon$-minimizers belong to $\mathscr C(K)$.
  As a consequence of this result we obtain the following sharp stability
inequality for crystalline norms: There exist $\gamma=\gamma(K,n)>0$ and
$\sigma=\sigma(K,n)>0$ such that, whenever $|E|=|K|$ and $|E\Delta K|\le
\sigma$ then
  $$
  \mathscr F(E) - \mathscr F(K^a)\ge \gamma |E \Delta K^a| \qquad \text{for
some }K^a \in \mathscr C(K).
  $$ In other words, the Wulff energy $\mathscr F$ grows very fast (with power
$1$) away from the set $\mathscr C(K).$ The set $K^a \in \mathscr C(K)$
appearing in the formula above can be informally thought as a sort of
"projection" of $E$ on the set $\mathscr C(K).$
  Another corollary of our result is a very strong rigidity result for
crystals: For crystalline surface tensions, minimizers of $\mathscr F(E)+\int_E
g$ with small mass are polyhedra with sides parallel to the ones of $K$. In
other words, for small mass, the potential energy cannot destroy the
crystalline structure of minimizers. This extends to arbitrary dimensions a
two-dimensional result obtained in [9].
