As part of a computational geometry question, I need a result for an intermediate step.
Suppose there is a convex polygon S, with finitely many points inside the polygon. Now, we want to find the line outside the polygon (i.e, the line can pass through a vertex or edge but cannot intersect) which minimizes the sum of perpendicular distances between the line and (vertices + interior points).
I have a hunch that we can prove that this line must be one of the edges, but I am unable to find a proof.
The simplest approach I tried was checking if rotating a line passing through vertex (but outside polygon) always decreases the sum till it reaches the edge. But this is clearly not true.
How to prove this?
Note 1: I think that this theorem must be true, but I have not seen it anywhere. It might be possible that it is actually false and the original question can be solved by a different method. If so, I would like to see a counterexample.
Note 2: The original question, the only answer proposes the theorem but does not provide a proof.