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 \[ W_\tau=\{{\mathbf x}\,{:}\; \langle\,{\mathbf x}|{\mathbf n}\,\rangle\leq \tau({\mathbf n})\,,\;\forall\,{\mathbf n}\in{\rm ext}W_\tau^*\}\,. \]