2. Prove that the divide-and-conquer algorithm for the closest-pair problem examines, for every point p in the vertical strip (see Figures 5.7a and 5.7b), no more than seven other points that can be closer to p than dmin, the minimum distance between two points encountered by the algorithm up to that point.

dy dr dmin FIGURE 5.7 (a) Idea of the divide-and-conquer algorithm for the closest-pair problem. (b) Rectangle that may contain points closer than dmin to point p. dy dr dmin FIGURE 5.7 (a) Idea of the divide-and-conquer algorithm for the closest-pair problem. (b) Rectangle that may contain points closer than dmin to point p