A strong form of the quantitative Wulff inequality for crystalline norms (Q6574279)

The notion of Wulff shape noted \(W_{\nu, f}=\{x\in \mathbb{R}^{n}: x.u\leqslant f(u)\), for all \(u\in \operatorname{supp} \nu \}\), where \(f: \mathcal{S}^{n-1} \rightarrow \left[0, \infty\right)\) and \(\nu\) is a Borel measure, has its origins in the classical theory of crystal growth. In more modern mathematical terms, it provides a unifying setting for several extremal problems with an underlying isotropic measure. The Wulff inequality plays an important role in several classical problems of the calculus of variations, geometric measure theory and mathematical physics. Several authors studied variational problems involving capacity, perimeter and measure [\textit{A. Figalli} et al., Inv. Math. 182, No 1, 167--211 (2010; Zbl 1196.49033); \textit{F. Alessio} and \textit{M. Francesco}, Arch. Ration. Mech. Anal. 201, No 1, 143--207 (2011; Zbl 1279.76005); \textit{N. Fusco} et al., Ann. Math. 168, No 3, 941--980 (2008; Zbl 1187.52009); \textit{R. Neumayer}, SIAM J. Math. Anal. 48, No 3, 1727--1772 (2016; Zbl 1337.49077); \textit{N. Fusco} and \textit{V. Julin}, Calc. Var. Partial Differ. Equ. 50, No 3--4, 925--937 (2013; Zbl 1296.49041); \textit{F. Alessio} and \textit{Y. R. Y. Zhang} Commun. Pure Appl. Math. 75, No 2, 422--446 (2022; Zbl 1495.49033); \textit{J. E. Taylor}, Bull. Am. Math. Soc., 84, No 4, 568--588 (1978; Zbl 0392.49022)].\N\NThe principal objective of this paper is to prove quantitative stability for crystalline anisotropic perimeters, with control on the oscillation of the boundary with respect to the corresponding Wulff shape, for \(n\geqslant 3\).