Question

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.

Answer 1

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