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I would like to prove that the minimal point of a intersection of $N$ convex sets in $\mathbb{R}^2$ is also the minimal point of the intersection of two of the aforementioned sets.

Rephrasing the statement for real functions of one variable: I would like to see that the minimum of a maximum of $N$ convex functions is the minimum of the maximum of two of the aforementioned functions.

Can this statement be proven?

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I'll use the functional formulation. Let $\varphi$ be the maximum of convex functions $\varphi_k$, $k=1,\dots,n$. Suppose $\varphi$ attains minimum at $x_0$. Recall that convex functions have one sided derivatives. It is not hard to prove that $\varphi'_+ = \max_k (\varphi_k)'_+$ and $\varphi'_- = \min_k (\varphi_k)'_-$. Since $x_0$ is a minimum, we have $\varphi'_+(x_0)\ge 0$ and $\varphi'_-(x_0)\le 0$. Thus, we can pick two indices $j,k$ such that $(\varphi_j)'_+(x_0)\ge 0$ and $(\varphi_k)'_-(x_0)\le 0$. The function $\max(\varphi_j,\varphi_k)$ attains its minimum at $x_0$.

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