Interface free energy/represent

There is a tangent plane to \(W_\tau\) at \({\mathbf x}\) iff \(W_\tau\) has a unique support plane \(\partial H({\mathbf n})\) containing \({\mathbf x}\ ;\) the outward normal to \(W_\tau\) at \({\mathbf x}\) is well-defined and equal to \({\mathbf n}\ .\) A tangent plane is always an extremal support plane. Indeed, suppose that \({\mathbf n}=\lambda{\mathbf n}_1+(1-\lambda){\mathbf n}_2\) with \(0<\lambda<1\) and \({\mathbf n}_1, {\mathbf n}_2\in {\rm ext}W^*_\tau\ ;\) suppose that \(\partial H({\mathbf n})\) is the unique support plane at \({\mathbf x}\in\partial W_\tau\ .\) Then \(\tau({\mathbf n})=\langle\,{\mathbf x}|{\mathbf n}\,\rangle\) and \[ 1=\tau({\mathbf n})=\lambda \langle\,{\mathbf x}|{\mathbf n}_1\,\rangle+(1-\lambda)\langle\,{\mathbf x}|{\mathbf n}_2\,\rangle\leq \lambda\tau({\mathbf n}_1)+(1-\lambda)\tau({\mathbf n}_2)=1\,. \] Therefore \[ \lambda\underbrace{(\tau({\mathbf n}_1)-\langle\,{\mathbf x}|{\mathbf n}_1\,\rangle)}_{\geq 0}+(1-\lambda)(\tau({\mathbf n}_2)-\langle\,{\mathbf x}|{\mathbf n}_2\,\rangle)=0\,, \] and \(\tau({\mathbf n}_i)=\langle\,{\mathbf x}|{\mathbf n}_i\,\rangle\) so that \(\partial H({\mathbf n}_i)\) is a support plane at \({\mathbf x}\ .\) Hence \({\mathbf n}={\mathbf n}_1={\mathbf n}_2\ ,\) the decomposition of \({\mathbf n}\) is trivial and \({\mathbf n}\) is an extremal point of \(W_\tau^*\ .\) We have extremal support planes which are not tangent planes when \(W_\tau\) has an edge or a corner.