Some Intuition about Fact 3

We can think of every orthogonal polygon as a rectangle with smaller rectangles cut out of corner.

If we start with a rectangle (n = 4), we have 4 convex corners and 0 concave corner. Each subsequent move of removing a smaller rectangle from the corner introduces exactly 1 more convex corner and 1 more concave corner.

Hence, the number of concave corners is 4 less than the number of convex corners for any orthogonal polygon P.