Problem: Consider a set of $n$ points in the plane, how could we find a strip of minimal vertical distance that contains all points?
Definitions: A strip is defined by two parallel lines and the vertical distance is defined as the distance between their intersection points with the $y$ axis.
3 variables solution: In the plane itself, this could be solved using a linear program of three variables, $m$, $a$ and $b$ where we look for $y=m\cdot x+a$ and $y=m\cdot x+b$.
Duality: If we move to the dual plane, we get a set on $n$ lines which can be transformed to $n$ upper half-planes or $n$ bottom half-planes. Denote $C_1$ to be the intersection of all upper half-planes intersection and $C_2$ of the bottom ones. The strip in the dual problem is represented by the two ends of the shortest vertical segment crossing the $C_1$ and $C_2$.
My question is - can we express the problem in the dual plane using a linear program of two variables?