# Interface free energy/show

If $$\tau({\mathbf u})\leq 1\ ,$$ it follows from $W_\tau:=\{{\mathbf x}\,{:}\; \langle\, {\mathbf x}|{\mathbf n}\,\rangle\leq \tau({\mathbf n})\,,\;\forall\, {\mathbf n}\}$ that $${\mathbf u}\in W^*_\tau\ .$$ Conversely, since $$\tau$$ is the support function of $$W_\tau\ ,$$ there exists for any $${\mathbf u}$$ a point $${\mathbf z}\in W_\tau$$ such that $$\langle\,{\mathbf z}|{\mathbf u}\,\rangle=\tau({\mathbf u})\ .$$ Therefore, when $${\mathbf u}\in W^*_\tau\ ,$$ we have $$\tau({\mathbf u})=\langle\,{\mathbf z}|{\mathbf u}\,\rangle\leq 1\ .$$ Hence $W^*_\tau=\{{\mathbf u}\,{:}\; \tau({\mathbf u})\leq 1\}\quad\text{and}\quad \tau({\mathbf x})=\min\{t\geq 0\,{:}\; {\mathbf x}/t\in W^*_\tau\}\,.$