I plotted 4 random points on the 2D plane and calculated the shortest and the longest paths joining all 4 points together. The image below shows the longest (blue) and the shortest (black) paths that connects all 4 points in a continuous line. There are totally 6 line segments connecting the 6 possible pairs of points. Only 3 line segments are needed to connect all 4 points. My simulation shows that there are no overlapping line segments for the two paths. Similar results are obtained every time.
Is there any prove to this and has anyone heard of it?