I faced a minimax problem with a vector and discrete set of functions as shown below. $$ \min_{\mathbf{x}\in \mathbb{R}^{+n}\\\sum_{k=1}^n\mathbf{x}_k=1} \max_{l\in \{1,2\cdots \lceil n/2 \rceil \}} f_{l}(\mathbf{x})$$
The following is known about the problem:
Cond 1. $f_l(\mathbf{x})$ is convex with respect to $\mathbf{x}$ and has a minimum at $\mathbf{x}^{(l)}$.
Cond 2. For all pairs of $l^\prime < l, f_l(\mathbf{x}^{(l)}) < f_{l^\prime}(\mathbf{x}^{(l^\prime)}) = f_{l}(\mathbf{x}^{(l^\prime)}) < f_{l^\prime}(\mathbf{x}^{(l)})=\infty. $ Moreover, the optimal points are in the constraint of $\min$.
Cond 3. If $\mathbf{x}=\sum_{l>l^\prime}a_{l}\mathbf{x}^{(l)}$, where $\sum_{l>l^\prime}a_{l}=1$, then $f_{l^\prime}(\mathbf{x})=\infty$.
I want to check whether the solution of the problem is $f_1(\mathbf{x}^{(1)})$.
If you can provide the answer, it would be greately appreciated. However, it would also be very helpful if you could check whether the following statements are true or false:
Check 1. Is the solution $\mathbf{x}^\star $ in $ \mathrm{span}(\{\mathbf{x}^{(1)} \cdots \mathbf{x}^{(\lceil n/2 \rceil)}\}) $?
Check 2. Is the solution $\mathbf{x}^\star$ located at an intersection of some pair of functions( $\exists (l^\prime, l)$ such that $f_l(\mathbf{x}^\star)=f_{l^\prime}(\mathbf{x}^\star)$) or $\mathbf{x}^\star\in\{\mathbf{x}^{(1)} \cdots \mathbf{x}^{(\lceil n/2 \rceil)}\}$?
I found my answer.
My argument turned out to be true.
Let $F(x):=\max_l f_l(x)$ Then, $$F(x)\ge f_1(x) \ge f(x^{(1)})$$ which implies $$\min_x F(x) \ge f(x^{(1)}).$$
Also, $$F(x^{(1)})=\max(\{ f_l(x^{(1)}) : l\in \{1,\cdots, d\}\})=\max(\{f_1(x^{(1)})\})=f_1(x^{(1)})$$
Therefore, $$\min_x F(x) = f(x^{(1)}).$$