I need to find an equation for the upper bound of $\max \mathbf{w}^T\mathbf{x}_i, \; i=1, \dots N$.
where $\mathbf{w}$ and $\mathbf{x}_i$ are two vectors.
I need to find a function $f$ which holds the following inequality.
$\max \mathbf{w}^T\mathbf{x}_i \leq \mathbf{w}^T \mathbf{z}$
where $\mathbf{z} = f(\mathbf{x}_i),\; i=1, \dots, N$
e.g
Let $\mathbf{x}_1 = \begin{pmatrix}x_{11}\\x_{12}\\x_{13}\end{pmatrix}, \; \dots, \mathbf{x}_N = \begin{pmatrix}x_{N1}\\x_{N2}\\x_{N3}\end{pmatrix}$
$\mathbf{z} = \begin{pmatrix}f(x_{11}, \dots, x_{N1})\\f(x_{12}, \dots, x_{N2})\\f(x_{13}, \dots, x_{N3})\end{pmatrix}$
for example f can be a $\max$ or $\min$ function.
All the values of $\mathbf{x}_i, \;, i=1, \dots, N$ are known. But $\mathbf{w}$ is unknown.
Is it possible to have $f$ as a function only on $\mathbf{x}_i$?
Example: $\mathbf{x}_1 = \begin{pmatrix}-10\\1\\3\end{pmatrix}, \; \mathbf{x}_2 = \begin{pmatrix}5\\-3\\-5\end{pmatrix} \implies \mathbf{z} = \max\mathbf{x}_i = \begin{pmatrix}5\\1\\3\end{pmatrix}$