The problem is that in many dimensions that often just won't work. The cloud of points can be smeared in directions that don't line up with any axes, in such a

The problem is that in many dimensions that often just won't work. The cloud of points can be smeared in directions that don't line up with any axes, in such a way that no matter which 2 axes you choose the clusters will overlap (but choosing a plane that doesn't align with any axis will clearly separate them). Even in 3 dimensions, you can see this problem by considering your populations to lie on 2 parallel planes, x+y+z=1 and x+y+z=2. Looking at this in x-y, x-z, or y-z projections, the point clouds will often be inseparable. And yet if you know where to look, all it takes is a projection to one dimension (f(x,y,z) = x+y+z) to separate them cleanly. The curse of dimensionality appears here in the fact that the "number" of possible ways of projecting the space increases dramatically with the dimension.