Answer to Question #238 in Geometry for Tony

Question

On the plane there are several points, all the pairwise distance between them are different. Each of these points connect with the nearest. Can this set be a closed broken line?

Answer 1

Suppose that we have closed polygon. Let AB be the largest element of the broken line, AC and BD be an adjacent links. Then AC < AB, i.e. B is not the closest to the point A, and BD < AB, i.e. A is not the closest to the point B. Hence the points A and B cannot be connected. A contradiction occurs.