# Wulff theorem

###### Definition 1 (Wulff shape).

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

 $W_{\phi}:=\{x\in\mathbb{R}^{n}\colon\text{\langle x,y\rangle\leq 1 for all % y such that \phi(y)\leq 1}\}$

(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, $1$-homogeneous function. Given a regular open set $D\subset\mathbb{R}^{n}$ we consider the following anisotropic surface energy:

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

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 $D$ with the same volume as $W_{\phi}$, i.e. $|D|=|W_{\phi}|$, one has $F_{\phi}(D)\geq F_{\phi}(W_{\phi})$. Moreover if $|D|=|W_{\phi}|$ and $F_{\phi}(D)=F_{\phi}(W_{\phi})$ then $D$ is a translation of $W_{\phi}$ i.e. there exists $v\in R^{n}$ such that $D=v+W_{\phi}$.

Title Wulff theorem WulffTheorem 2013-03-22 15:19:50 2013-03-22 15:19:50 paolini (1187) paolini (1187) 8 paolini (1187) Theorem msc 52A21 FinslerGeometry Wulff shape