Minimal point of a intersection of $n$ convex sets is always the minimal point of the intersection of two convex sets?

Question

I have the intuition that the minimal point (in the sense of having the lowest value in one of the Euclidean space dimensions) of a intersection of $n$ convex sets is always the minimal point of the intersection of two convex sets in the family. Has this been proven before?

Answer 1

In $d$ dimensions, you need at least $d$ convex sets, not two. For example, in a suitably rotated polytope in $d$ dimensions, the minimal point will be a vertex, which is defined by the intersection of $d$ planes.