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Function Euclidean Distance Convex Set Volume Surface Area Inner Product Inner Automorphism Affine Transformation T N B Frame Coercive Function Regular Open Set

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Wulff Theorem

Definition 1 (Wulff shape) Let $\phi\colon\mathbb{R}^n \to [0,+\infty)$ be a non-negative, convex, coercive, positively -homogeneous function. We define the Wulff shape relative to $\phi$ as the set

$\displaystyle W_\phi := \{ x \in \mathbb{R}^n \colon$ $\langle x,y\rangle \le 1$ for all

such that $\phi(y)\le 1$ $\displaystyle \}$

(where $\langle \cdot,\cdot\rangle$ is the Euclidean inner product in $\mathbb{R}^n$ .)

Theorem 1 (Wulff) Let $\phi\colon\mathbb{R}^n \to [0,+\infty)$ be a non-negative, convex, coercive, -homogeneous function. Given a regular open set $D\subset\mathbb{R}^n$ we consider the following anisotropic surface energy:

$\displaystyle F_\phi(D) = \int_{\partial D} \phi(\nu_D(x))\, d\sigma(x)$

where $\nu_D(x)$ is the outer unit normal to $\partial D$ , and $\sigma$ is the surface area on $\partial D$ . Then, given any set with the same volume as $W_\phi$ , i.e. $\vert D\vert=\vert W_\phi\vert$ , one has $F_\phi(D)\ge F_\phi(W_\phi)$ . Moreover if $\vert D\vert=\vert W_\phi\vert$ and $F_\phi(D)=F_\phi(W_\phi)$ then is a translation of $W_\phi$ i.e. there exists $v\in R^n$ such that $D=v+W_\phi$ .

1	Convex Set
1	Euclidean Distance
1	Function
2	Volume
2	Area
2	Inner Product
2	Surface
3	Affine Transformation
3	Inner Automorphism
8	T N B Frame
13	Regular Open Set
15	Coercive Function