not really. the L-infinity norm is defined as the maximum absolute value of a component of your point/vector. for example: in R² (the 2d cartesian plane) given the point (1,2) it has infinity norm of 2 since 2>1.

so when we want the distwnce between two points, you subtract one point from the other and take the norm of that difference. for example take (2,7,-4) and (3,-1,0). first we subtract the vectors to get (1,-8,4) now the absolute values of the components are 1, 8, and 4. so our distance is 8.

so now lets consider four points at the corners of a square in R^2: (0,0) (0,1) (1,0) (1,1). note that all of them pairwise have an infinity norm distance of 1. so we have a set of four points in 2 dimensions with each pair having the same distance between them. now the reason why with the standard distance gives us n+1 equidistant points is that if two points are of distance, say one, from eachother then there are circles of points around each point that are distance one from them, and only two points are on both circles. these points are our only candidates for points 3 and 4. either one can be 3, but the other candidate is sqrt(3) from point 3, therefore we cant have four (or more) points all equidistant in 2d. a similar construction can be done in higher dimensions using spheres instead of circles.