A Strong Form of the Quantitative Wulff Inequality for Crystalline Norms

Abstract

Quantitative stability for crystalline anisotropic perimeters, with control on the oscillation of the boundary with respect to the corresponding Wulff shape, is proven for $n\geq 3$. This extends a result of [Neu16] in $n=2$.

Figures (2)

• Figure 1: Left: A parallel perturbation in the direction $\nu_i$, with $F_j^{\bf a}\Delta F_j^{\bf a'}$ emphasized. Right: The set $F_j^{\bf a}\Delta F_j^{\bf a'}$ is contained within a slab $S$ of height $|{\bf a_{\it{i}}}-{\bf a_{\it{i}}'}|$ and the plane $\Sigma_{\nu_j}^{1+{\bf a_{\it{j}}}}$. As $F_j^{\bf a}\Delta F_j^{\bf a'}$ is slanted (according to $\nu_j$), its height is bounded by $\csc(\theta_{ij})\cdot {\bf d_{\it{i}}}|{\bf a_{\it{i}}}-{\bf a_{\it{i}}'}|$.
• Figure 2: Left: $K, K^{\bf a},$ and a particular $rK$ with $0<r<1$. Right: We can bound $rF_i\cap (K\setminus K^{\bf a})$ in terms of the contributions between the parallel hyperplanes $S_{\nu_j}^{1+{\bf a_{\it{j}}},r}$. In this case, $\mathscr{I}_i^{\bf a}(r)=\{j_1,j_2\}$.

Theorems & Definitions (40)

• Theorem 1.1
• Theorem A: FZ2019
• Remark 2.1
• Lemma 2.2: Neumayer2016
• Proposition 2.3: Neumayer2016
• Lemma 2.4: Neumayer2016
• Lemma 2.5: Neumayer2016
• Lemma 2.6: Neumayer2016
• Lemma 2.7: Neumayer2016
• Remark 2.8