> One big problem with the conclusion is that intuitions from low dimensional spaces often don’t carry over to high dimensional spaces. e.g. common example is how the volume of the unit hypersphere intersected with hypercube ratio goes to zero. One funny thing I saw once was something like “the real curse of dimensionality is how different optimization in high dimensional spaces is compared to low dimensional spaces”.

How so usually in Pure Mathematics(Analytic area's) everything done in R^{1} R^{2} is usually generalized to R^{N} and much of the intuition carriers over. So how does it fail here in the context of Machine Learning ?

Note: I know nothing about ML :(