Strong stability for the Wulff inequality with a crystalline norm

Let $K$ be a convex polyhedron and $\mathcal{F}$ its Wulff energy, and let $\mathcal{C}(K)$ denote the set of convex polyhedra close to $K$ whose faces are parallel to those of $K$. We show that, for sufficiently small $ϵ$, all $ϵ$-minimizers belong to $\mathcal{C}(K)$. 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 $\mathrm{\mid }E\mathrm{\mid }=\mathrm{\mid }K\mathrm{\mid }$ and $|E\Delta K|\le \sigma$then$$ \mathscr F(E) - \mathscr F(K^a)\ge \gamma |E \Delta K^a| \qquad \text{for some }K^a \in \mathscr C(K). $$Inotherwords,theWulffenergy$\mathscr F$growsveryfast(withpower$1$)awayfromtheset$\mathscr C(K).$Theset$K^a \in \mathscr C(K)$ appearing in the formula above can be informally thought as a sort of “projection” of $E$ on the set $\mathcal{C}(K)\mathrm{.}$ Another corollary of our result is a very strong rigidity result for crystals: For crystalline surface tensions, minimizers of $\mathscr F(E)+\int_E g$withsmallmassarepolyhedrawithsidesparalleltotheonesof$K$. 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].