Let $\mathcal{S}\in\mathbb{R}^n$ be a convex, bounded surface. Now consider the rescaling operator,
\begin{equation} \mathcal{O}_t(\mathcal{S})=\{(tx_1,\dots, tx_n)\in\mathbb{R}^n | (x_1,\dots,x_n)\in\mathcal{S} \}, \ \ t\in[0,\infty). \end{equation} How would you set up the following proof (if the statement is true at all): let $(x_1^*,...,x_n^*)\in\mathbb{R}^n$, then there exists a unique $t^*$ such that $(x_1^*,...,x_n^*)$ lies on the surface $\mathcal{O}_{t^*}(\mathcal{S})$.