Suppose S and A show a square and a set involves the points inside the S, respectively. Moreover, B is a set that involves points outside the S. Let E is a subset of A that includes edge/boundary points. Actually, the edge points are the nearest points in A to the points in B.
I want to know, how does the E or a another subset of A that includes E can be determined based just on A without exhaustive search on A. I mean, does exist a geometric solution for this. For example, maybe it's possible to find a circle with radius r inside the square S that involves the points set named C. Obviously C is a subset of A. So, we can say that E is a subset of A-C.
Thank you very much for your consideration. I mean all the points that are located in the square. Suppose that we have a set S that includes some points in Euclidean space. Imagine an square in this space the divided all the points into two subsets A and B, where A is consisted of points that are located in the square and B is the complement of A (B=S-A). Now, for every points in B, we want to find the nearest point from A without exhaustive search. It is clear that for finding the exact nearest points, a comparison process among all the points in A and B should be done (exhaustive search). But perhaps it is possible to find a subset of A that certainly at least includes the nearest points.
I do not understand your idea. I think your thought about my question is wrong. The following figure describes my question.