Consider the two convex sets K₁ and K₂ pictured above. Each has an origin, namely and , and a support function and . Observe the two support lines that intersect at the point P. We are only going to consider convex sets which do not have straight line segments or corners, so that we can assume that for any point P that is outside of a convex set K, there are exactly two support lines to K that pass through P. We shall call these points of contact A₁ and A₂, called contact points or support points, and we shall denote the foot of the perpendicular from the origin to the support line by H₁ and H₂. Let (x₁, y₁) and (x₂, y₂) denote the coordinates of the origins and , respectively. Let represent the direction of the support line. Thus, the equations for the support lines are

		(1)

in terms of and . Differentiating the above equation, we get

		(2)

Note, also, that the length of the line segment from the point P to H_i can be expressed as follows:

		(3)

Recall that (try not to get confused with the notation; i.e., before we said HP=p', but here the point A represents the same point as P did before), and so it follows that

		(4)

Let t_i represent the distance of the line segment PA_i. Upon inserting equations (3) and (4) into equation (2), we get

		(5)

It must be noted that the distances mentioned so far are taken as positive if measured from P out towards H (or A), and taken negatively if measured from H (or A) towards P. Observe that the segment H₂A₂ is negative. Now, we proceed to take the exterior product of equation (5):

		(6)

or, it can also be expressed as

		(7)

A keen observation is that the angle also represents the angle made by the support lines at the point P. Now, if we integrate the left side of equation (7) for all values of and , we get

		(8)

Now let's consider when we try to integrate the right side of equation (7). What might be the consequences of only considering the two support lines on top of the respective convex sets like we did at the beginning? Are there not more support lines that should be considered? Yes there are. For any point P, there are two support lines to both K₁ and K₂ passing through P. Consider the figure below:

We have let denote the distances from P to the contact points, respectively, and also let denote the direction of the respective support lines. Thus, to integrate the right side of equation (7), we need to consider all possible combinations of the support lines. In addition, we will need to consider the angles made by each of these pairs of support lines; namely, in blue, in pink, in red, in green. Now, our integral takes the form

		(9)

A special case would be when , shown below; the formula above simplifies to

		(10)

The interesting thing about the last two formulas is that they are true for any convex set in the plane!

Problem 15: Consider the last diagram. Suppose the boundary of K has radius of curvature , where is the radius of curvature at and . It is easy to see that , where we define as the arc of K at each support point. We can rewrite Equation (7) to see that or Show that: (a) (b) Solution