There is this one property I am trying to prove(and not able to) for convex hulls of a finite set S of bounded points in two dimensions.
Property: Every point on the boundary of the convex hull lies on a line-segment joining two points p, q such that p, q belong to S.
A boundary point is defined as follows: A point p is a boundary point if for a small circle drawn around p, there will exist points both inside and outside the convex hull in the circle.
I am able to show the following: Lets say for a pair of points p and q, I can draw a line passing through them such that all the points of S either lie on the line or one particular side of the line. The half plane made by the line containing all the points will now be convex. So now we can argue that a point x on the open segment joinging p, q is a boundary point, since the points on the other side of the line are not part of the hull and the points on the line just beside x are on the hull. How do I show that every boundary point is like this?