The more interesting case is the Dobkin and Snyder algorithm. Its correctness relies on the following two statements:
A polygon is unimodal iff for every point x on its boundary a traversal of the entire boundary starting at x takes us monotonically further from x and then monotonically closer to x until we arrive back at x. By monotonically further from x, we mean that our distance from x may remain the same during portions of our traversal but it must never decrease; by monotonically closer, we mean the same except that the distance must never increase. For example, in Figure 13, the left polygon is unimodal with respect to point x because as we traverse the boundary from x to v by way of u, our distance from x continually increases. Then, traversing back to x by way of w our distance from x continually decreases. The right polygon is not unimodal with respect to point x because u and w are further from x than point v between them.
A convex polygon P = (p_{1}, p_{2}, p_{3}, p_{4}) of four points is sufficient to show that not all convex polygons are unimodal. See Figure 14.
The distance between p_{1} and p_{2} and between p_{1} and p_{4} is 1. The interior angle at p_{1} is greater than 0 and less than 180 degrees. The length of line segment p_{1}p_{3} is slightly less than 1 but long enough to intersect p_{2}p_{4}. As a result, p_{1} has two local maxima: p_{2} and p_{4}. We can generalize this example to show that, in fact, a point can have n local maxima in a convex polygon described by 2n points. These polygons were originally described in [1]. The distance between every point p_{i}, i even, and p_{1} is 1. The length of line segment p_{1}p_{i}, i odd, is slightly less than 1 but long enought to intersect line segment p_{i - 1}p_{i + 1}. Thus, all points p_{i}, i even, are local maxima for point p_{1}. The resulting convex polygon for ten points is illustrated in Figure 14.Earlier, we mentioned that the Dobkin and Snyder algorithm actually relies on the assumption that a local maximum is the maximum. In other words, if v is a local maximum of u then v is the furthest point from u. In the polygons just exhibited, this is actually true. However, if we perturb p_{2n} so that the distance between p_{1} and p_{2n} becomes slightly larger than 1 then the polygon remains convex but local maximum p_{2} is now closer to p_{1} than local maximum p_{2n}.