Minimal point of the intersection of convex sets.

I am trying to find out if there is any known result in convex optimization that implies the following statement: "A minimal point of the intersection of $N > 2$ convex sets in $\mathbb{R}^2$ is a minimal point of the intersection of $2$ sets among the aforementioned $N$."
and more generally: "A minimal point of the intersection of $N >d$ convex sets in $\mathbb{R}^d$ is a minimal point of the intersection of $d$ sets among the aforementioned $N$."
Note: Minimal point is considered here to be a point in the set that attains the lowest value in one of the Euclidean space dimensions.

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I don't understand your "Note"--it makes no sense to me. – Włodzimierz Holsztyński Feb 12 '13 at 4:38
I'd like to dwell on your question, but I need a complete, precise definition of "minimal point", as you use it. – Włodzimierz Holsztyński Feb 12 '13 at 4:40