Problem 17: Density for Sets of Strips

Density for Sets of Strips

Consider the strip of breadth a depicted above. It is defined as the closed part of the plane determined by all points that lie between two parallel lines located a distance a from each other. We will deonte the strip by the letter B. Observe that the position of the strip B can be determined by the coordinates with respect to the midparallel of B (the dashed line). Thus, the density for sets of strips of fixed breadth a is

(1)

Problem 17 :

(a) Let K be a bounded convex set in the plane. Prove that if and only if the midparallel of B interesects the parallel set Ka/2 of K.

(b) Show that the measure of the set of strips of breadth a that intersect a convex set K is .

Special Cases

The measure of the set of strips having breadth a that contain a point P is

(2)
The measure of the set of strips having breadth a that contain a line segment S of length s is
    (3)

    Now consider the set of strips that contain a given set. To make things simple, let's assume that the set is convex and its diameter . In this case, for a strip to contain a given set, the boundary of the strip must not meet K. Thus, this measure is the difference between the measure of the set of strips of breadth a that intersect a convex set K and twice the measure of the set of lines that intersect the set K (why?); that is,

    (4)

      Problem 17 (c):

      Show that the measure above is not negative.

      Solution

      In terms of geometric probability, let K1 be a convex set that is contained within a bounded convex set K. The probability that a strip of breadth a dropped randomly onto K intersects K1 is given by

    (5)
    where L1 and L are the perimeters of K1 and K, respectively.
    In the case that the diameter of K1 is less that or equal to a, then the probability that the strip B contains K1 is
    (6)

    Now consider N convex sets Ki, for i=1,2,...,N, that are located in a bounded convex set K, as pictured in the diagram below. Let L denote the perimeter of K and let Li denote the perimeter of Ki. Let n represent the number of sets Ki that are intersected by the srtip B; thus, we have

      (7)

      In the case that all the diameters of Ki are less than or equal to a, for i=1,2,...,N, then let ni denote the number of sets Ki that are contained within the strip B. Thus, integrating over all strips that intersect K, we have

        (8)

        Now we can make the following statement:

        Let Ki (i=1,2,...,N) be N convex sets contained within a bounded convex set K. The mean number of sets Ki that are intersected by a randomly dropped strip B of breadth a is given by
        (9)
        And if the diameters of the sets Ki are all less than or equal to a, then the mean number of sets Ki that are contained within B is given by
        (10)




        Sets of Points, Lines, and Strips

        Let D be a domain in the plane with area F. Note D is not necessarily convex as shown in the diagram below. The denisty for sets of pairs of points and strips (P,B), provided that P and B are independent of each other, is . Thus, the measure of the set of pairs (P,B) such that the set is given by

        (11)

        In order to calculate this intergral, let us fix P and incorporate equation (2). Thus, we have

        (12)

        where, as usual, a is the breadth of B. Now, if we come from the other direction, we fix B and denote the area of by f, we get

        (13)

        and thus it follows that

        (14)

        Now consider the convex hull of the domain D, and denote its perimeter by L. Observe that and , and thus we can make the following statement:

        Given that P and B are chosen at random under the conditions that and , the probability that is given by
        (15)

        and the mean value for the area f of the intersection of B and D is given by

        (16)

        Suppose K is a convex set and consider the sets of pairs of lines and strips (G,B) such that . The measure of this set is given by the integral of the differential form over the set . To evaluate this integral, either fix G and then integrate over the strips B or fix B and then integrate over the lines G. Using the former method, we have

          (17)

          where is the length of the chord . The latter method yields

          (18)

          where u represents the perimeter of . Thus, from equations (17) and (18), if follows that

          (19)

          In terms of geometric probability, the preceding results can be stated as the following:

          Suppose that G is a line and B is a strip of breadth a, and they are chosen at random such that . Thus, the probablity that is given by
          (20)
          If a=0, the previous result reduces to the case that two random chords of K will intersect inside K; that is
          (21)
          The mean length of the boundary of is thus given by
          (22)

          Now, let us consider the density for pairs of strips , which is . Using the result from Problem 17 (a) and Equation (18), we have that the measure of the set of pairs of strips such that is given by

          (23)

          where, a1 and a2 denote the breaths of strips B1 and B2, respecitvely.

          Therefore, we can state that

          If are two random strips intersecting the convex set K, the probability that B1 and B2 also intersect each other within the set K is given by

          (24)

          References: Santalo