Let’s begin with following definitions from Wikipedia:
A convex polygon is a simple polygon (not self-intersecting) in which no line segment between two points on the boundary ever goes outside the polygon.
A concave polygon is a simple polygon (not self-intersecting) which has at least one reflex interior angle – that is, an angle with a measure that is between 180 degrees and 360 degrees exclusive.
We know that 3 sided polygons (a.k.a triangles) are always convex. So, they are not very interesting. Let’s define the following procedure:
Midpoint Polygon Procedure: Given a n-sided simple polygon, join the consecutive midpoints of the sides to generate another n-sided simple polygon.
Here are few examples:
We observe that if the original polygon was a concave polygon then the polygon generated by joining midpoints need not be a convex polygon. So, let’s try to observe what happens when we apply the midpoint polygon procedure iteratively on the 4th case above (i.e. when we didn’t get a convex polygon):
Based on this observation, I want to make following conjecture:
Midpoint Polygon Conjecture: For every simple polygon there exists a smallest natural number such that after iterations of midpoint polygon procedure, we obtain a convex polygon.
If the given polygon is convex, then there is nothing to prove since , but I have no idea about how to prove it for concave polygons. I tried to Google for the answer, but couldn’t find anything similar. So, if you know the proof or counterexample of this conjecture, please let me know.