Comment by iNic

Yes that is true, but there are other ways to see spiky-like behavior.

First, the volume of spheres (or balls rather) in higher dimensions goes to zero as the dimension grows. Said another way, to keep unit volume on a ball you need to grow the radius more and more (which I interpret as spiky).

Second, the volume of spherical caps grows like ~exp(- d h^2 /2), in particular the caps lose volume fast in higher dimensions. To interpret this as "spikyness" I like to visualize it as two balls intersecting (which is just 2x the cap volume). If they are of the same radius, but their centers are just slightly off their intersection volume goes to zero quickly!