It really depends on what we assume about "natural" data. If it looks "positively curved", e.g. the uniform measure on the boundary of a convex body, or the a g

It really depends on what we assume about "natural" data. If it looks "positively curved", e.g. the uniform measure on the boundary of a convex body, or the a gaussian, or something, this holds. But if the distribution exhibits a strong hierarchical structure that's not so good. I think it's a plausible if not obviously true assumption.