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 \[ W^*_\tau=\{{\mathbf u}\,{:}\; \tau({\mathbf u})\leq 1\}\quad\text{and}\quad \tau({\mathbf x})=\min\{t\geq 0\,{:}\; {\mathbf x}/t\in W^*_\tau\}\,. \]