# 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.