# Interface free energy/label

For any $${\mathbf n}$$ let $$\phi_{\mathbf n}$$ be the affine functional ${\mathbf x}\mapsto \phi_{\mathbf n}({\mathbf x}):=\langle\,{\mathbf x}|{\mathbf y}\,\rangle-\tau({\mathbf y})$ so that $$H({\mathbf n})=\{{\mathbf x}\,{:}\; \phi_{\mathbf n}({\mathbf x})\leq 0\}\ .$$ The boundary of the half-space $$H({\mathbf n})$$ is the plane $$\partial H({\mathbf n})=\{{\mathbf x}\,{:}\; \langle\, {\mathbf x}|{\mathbf n}\,\rangle= \tau({\mathbf n})\}\ .$$ The equilibrium shape can be written as $W_\tau=\bigcap\{H({\mathbf n})\,\colon\;\|{\mathbf n}\|=1\}\,.$ A plane $$\partial H({\mathbf n})$$ is an extremal support plane of $$W_\tau$$ iff $$\phi_{{\mathbf n} }({\mathbf x})$$ cannot be written as $\phi_{{\mathbf n} }({\mathbf x})=c_1\phi_{{\mathbf n}_1}({\mathbf x})+c_2\phi_{{\mathbf n}_2}({\mathbf x})\quad c_1>0\,,\;c_2>0\,,$ except by taking $${\mathbf n_1}=t_1{\mathbf n}$$ and $${\mathbf n}_2=t_2{\mathbf n}\ ,$$ $$t_1>0$$ and $$t_2>0\ .$$ Let $$H({\mathbf n}_1)\not=H({\mathbf n}_2)\ ,$$ $$c_1>0$$ and $$c_2>0$$ be given. Let $${\mathbf n}=c_1{\mathbf n}_1+c_2{\mathbf n}_2$$ and $${\mathbf x}\in W_\tau\ .$$ Then, by sublinearity of $$\tau\ ,$$ $0\geq \phi_{{\mathbf n} }({\mathbf x})=\langle\,{\mathbf x}|{c_1{\mathbf n}_1+c_2{\mathbf n}_2}\,\rangle-\tau(c_1{\mathbf n}_1+c_2{\mathbf n}_2) \geq c_1\phi_{{\mathbf n}_1}({\mathbf x})+c_2\phi_{{\mathbf n}_2}({\mathbf x})\,.$ From this one gets that $$\partial H({\mathbf n})$$ is extremal iff $\tau({\mathbf n})<c_1 \tau({\mathbf n}_1)+c_2 \tau({\mathbf n}_2)\quad\forall\,{\mathbf n}_1,\,{\mathbf n}_2\; \text{linearly independent, such that}\; c_1{\mathbf n}_1+c_2{\mathbf n}_2={\mathbf n}\,,$ that is, iff the interface perpendicular to $${\mathbf n}$$ is stable (see definition). When the support planes $$\partial H({\mathbf n})$$ are parametrized by $${\mathbf n}\in \partial W^*_\tau\ ,$$ $$\partial H({\mathbf n})$$ is extremal iff $${\mathbf n}$$ is an extremal point of $$W^*_\tau\ .$$ Indeed, if $${\mathbf n}=\lambda{\mathbf n}_1+(1-\lambda){\mathbf n}_2$$ is a non-extremal boundary point of $$W^*_\tau\ ,$$ then $1=\tau({\mathbf n})\leq \lambda\tau({\mathbf n}_1)+(1-\lambda)\tau({\mathbf n}_2)\leq 1\;\implies\; \tau({\mathbf n})=\lambda\tau({\mathbf n}_1)+(1-\lambda)\tau({\mathbf n}_2)\,,$ so that $$\partial H({\mathbf n})$$ is non-extremal. Conversely, if $$\partial H({\mathbf n})$$ is non-extremal, then there exist $$c_1>0\ ,$$ $$c_2>0\ ,$$ $${\mathbf n}_1$$ and $${\mathbf n}_2$$ so that $${\mathbf n}=c_1{\mathbf n}_1+c_2{\mathbf n}_2$$ and $$\tau({\mathbf n})=c_1 \tau({\mathbf n}_1)+c_2 \tau({\mathbf n}_2)\ .$$ Putting $${\mathbf u}={\mathbf n}/\tau({\mathbf n})\ ,$$ $${\mathbf u}_1={\mathbf n}_1/\tau({\mathbf n}_1)$$ and $${\mathbf u}_2={\mathbf n}_2/\tau({\mathbf n}_2)\ ,$$ we get ${\mathbf u}= \frac{c_1\tau({\mathbf n}_1)}{\tau({\mathbf n})}{\mathbf u}_1+\frac{c_2\tau({\mathbf n}_2)}{\tau({\mathbf n})}{\mathbf u}_2\quad \text{and}\quad\frac{c_1\tau({\mathbf n}_1)}{\tau({\mathbf n})}+\frac{c_2\tau({\mathbf n}_2)}{\tau({\mathbf n})}=1\,,$ so that $${\mathbf u}\in\partial W^*_\tau$$ is non-extremal. To summarize, the support plane $$\partial H({\mathbf n})$$ of $$W_\tau\ ,$$ with $${\mathbf n}\in \partial W^*_\tau\ ,$$ is extremal iff $${\mathbf n}$$ is an extremal point of $$W^*_\tau\ .$$ This happens iff the interface perpendicular to $${\mathbf n}$$ is stable. The equilibrium shape can be rewritten as $W_\tau=\{{\mathbf x}\,{:}\; \langle\,{\mathbf x}|{\mathbf n}\,\rangle\leq \tau({\mathbf n})\,,\;\forall\,{\mathbf n}\in{\rm ext}W_\tau^*\}\,.$