It appears that the author makes a mistake in his attempts to simplify the problem, because although he is correct that he only needs to look at points on the e

It appears that the author makes a mistake in his attempts to simplify the problem, because although he is correct that he only needs to look at points on the edges, he goes on to suggest that he is looking only at corners of the polygon, and not at any of the (infinite number of) points between the corners.