This is a random curiosity I had but couldn't solve, let's say I have a shape in 2D or 3D, and add the distances from all vertices. For example, the square bellow would add up to $5+5+5+5+5\sqrt{2}+5\sqrt{2} = 20 + 10\sqrt{2}$, can I find a different shape with the same number of vertices and the same sum of the distances? On this example would be another quadrilateral that also adds up to $20 + 10\sqrt{2}$
