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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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