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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This seems to be an immediate consequence of Helly's theorem, applied to the given convex sets and the set of points where the spacedimension in the "Note" is smaller than its minimum over the intersection. 

