About exchanging min and max and correctness of an inequality Let $v_i \in \mathbb{R}^{n}, \ i=1, \ldots, m, \ \  $ $\mathcal{S}$ a convex polyhedron and $x \in \mathbb{R}^{n}$ be given. Consider the following solution $(s^{*},i^{*})$ to the problem 
\begin{equation}
\underset{s \in \mathcal{S}}{\max} \underset{j=1, \ldots, m}{\min} \langle v_j, s-x\rangle
\end{equation} 
I would like to conclude the following  Claim: 
Let $i \in \{1, \ldots, m\}$ be arbitrary, then it holds that
\begin{equation}
\langle v_i, s^{*}-x \rangle \geq \langle v_{i^{*}}, s^{*}-x \rangle \ . 
\end{equation} 
Question 1: Is the claim true?
Motivation: Why I think it might not be true: Consider first maximizing with repsect to $s \in S$, giving us an optimal $s$ for each $j=1, \ldots, m$, which we denote by $s^{(j)}$. Then minimzing with respect to $j$ the expressions $\langle v_j, s^{(j)}-x \rangle$, and denoting with $j^{(*)}$  the index where the minimum is attained. It is clear, that then 
\begin{equation}
\langle v_i, s^{(j^{*})}-x \rangle \geq \langle v_{j^{*}}, s^{(j^{*})}-x \rangle
\end{equation} 
Does not hold for arbitrary $i$, indeed it would only hold if we replace $s^{(j^{*})}$ with $s^{(i)}$ on the left hand side. So it is clear that if it is possible to exchange $\max$ and $\min$ in our expression, then the claim is false. This motivates my second question
Question 2 Can we exchange $\max$ and $\min$ ans still get the same solution to the optimization problem?
 A: Let $S:=\mathcal{S}$ and $a\cdot b:=\langle a,b \rangle$. Without loss of generality $x=0$ (or replace $S$ by $S-x$). That $(s^*,i^*)$ is a solution to the max-min optimization problem means the following: 
$\forall s\in S$ $\exists i_s\in[m]:=\{1,\dots,m\}$ $\forall i\in[m]$
\begin{equation}
 v_i\cdot s\ge v_{i_s}\cdot s \tag{1}
\end{equation}
and $\forall s\in S$
\begin{equation}
 v_{i^*}\cdot s^*\ge v_{i_s}\cdot s, \tag{2}
\end{equation}
where $i^*:=i_{s^*}$. 
So, (1) immediately implies 
\begin{equation}
 v_i\cdot s^*\ge v_{i_{s^*}}\cdot s^*= v_{i^*}\cdot s^*,
\end{equation}
which answers your Question 1 affirmatively. 
As for Question 2, selecting $S$ and $v_i$'s at random easily provides a counterexample, resulting in the answer "No" to Question 2. E.g., let $n=2$, $x=0\in\mathbb R^2$, $S:=\{(z,w)\in\mathbb R^2\colon z\ge-10,w\ge-10,z+w\le20\}$, $v_1=(3,9)$, $v_2=(8,2)$. Then 
\begin{equation}
 \max_{s\in S}\min_{j\in[m]} v_j\cdot s=110<220=\min_{j\in[m]}\max_{s\in S} v_j\cdot s. 
\end{equation}
