the volume of an n-ball peaks at n=4 dimensions and quickly drops to zero around n=20. cf. https://news.ycombinator.com/item?id=3995930

The comment refers to lebesgue measure (I don't even what), but I'd intuitively and ignorantly assume we count all faces of all n-1 balls (recursively) whereas the Volumes overlap and so the total (in lebesgue ...space?) is less than the sum of it's parts (in euclidean space) - how far off am I? (will delete if too far)