External Cover: The external cover Ce of K1 and K2 is the boundary of the convex hull of . Note that the convex hull of a set of points in the plane is defined as the intersection of all convex sets that contain this set of points.

If you have a group of convex sets, think of the convex hull as a the region created by a rubber band that is stretched around all of the sets.

You can think of the external cover as a rubber band placed around the two sets, where the length of the rubber band is denoted L_e. In the same fashion, and assuming that K₁ and K₂ do not overlap, let the internal cover C_i be a rubber band placed around the two sets so that it crosses at a point O in between them. We will denote its perimter by L_i. These are illustrated in the figure below.

For reasons that are not yet obvious, we will consider as separate curves, although some of their arcs overlap. Disregarding support lines because they belong to a set of measure zero, every line that intersects both K₁ and K₂ has two intersection points with the three curves , shown in blue, green and purple respecively and four intersection points with the curve , illustrated by the black points. Thus, there are a total of 10 intersection points. Consequently, we will denote the measure for the set of these lines by m₁₀.

Now we consider the set of lines that intersect either only K₁ or only K₂. These lines have a total of 6 points where they intersect the curves . Thus, m₆ denotes the measure of the set of these lines. The lines that separate the sets K₁ and K₂ have four points of intersection; namely, two with and two with . Similarly, we will denote the measure of the set of these lines by m₄.

Earlier in Problem 11, we saw that the measure of the set of lines that intersect a convex set is equal to the perimeter of that set. Thus, the measure of the set of lines intersecting K₁ while not intersecting K₂, denoted m₆', and the set of lines intersecting K₂ and not K₁, denoted m₆'', are as follows:

			(1)

and so the total of m₆ is

		(2)

Due to the fact that C_e itself is also a closed curve, it follows that

		(3)

Again, recall from Problem 11 that there is result in geometric probability that states that integrating the number of times a line intersects a convex curve over all lines that intersect this curve yields twice the perimeter. In other words,

		(4)

Applying this idea to the four curves we have considered, we see that

		(5)

You don't really have to understand how equation (5) comes from equation (4), just realize that there is a justification for it. Finally, we can determine m₄, m₆', m₆'', and m_{10 in terms of their respective perimeters. Thus, we have}

		(6)

_{So, we have a measure for the set of lines that (a) intersect both} K₁ and K₂, (b) intersect K₁ and not the other K₂, and (c) vice versa, and (d) that separate K₁ and K₂.

Now, suppose K₁ and K₂ are two bounded, convex, disjoint sets, and suppose that G is a random line that is under the condition of having to intersect the convex hull of K₁ and K₂. Then we can determine the following probabilities:

The probability that G intersects K₁ and K₂ is:

The probability that G intersects K₁ without intersecting K₂ is:

The probability that G intersects K₂ without intersecting K₁ is:

The probability that G separates K₁ and K₂ is:

Problem 13: Suppose we have two convex sets and such that . (a) What is the probability that a random line that intersects does not intersect ? (b) What is the probability that n random lines that intersect do not intersect ? Solution

References: Santalo; De-lin