It is well known that given points (x, y) and you need to calculate the Manhattan distances between them, instead of using |x1-x2|+|y1-y2| you can first convert all points (x, y) into (x+y, x-y) and the distances will become max(|x1-x2|, |y1-y2|) (also known as Chebyshev distance). Can this trick apply to higher dimensions?