There are several ways to leverage one's intuition from low-dimensional geometry to understand high-dimensional phenomena. For example, one can get a clearer picture of the behaviour of high-dimensional spheres by picturing them as being 'spikey'.
This excellent thread dealt with many such intutive ideas. One might think of the 'spikey spheres' visualisation as a 'bottom up' approach to high-dimensional geometry. We take a low dimensional object and change some of its properties so as to mimic its counterpart in high dimensions.
I've been wondering if there might be a dual 'top down' approach. Can we leverage ideas from infinite-dimensional analysis to understand high-dimensional geometry? Here's a more concrete question to get the ball rolling. It is known that there is no locally finite translation-invariant measure on an infinite-dimensional Banach space. This is clearly a purely infinite-dimensional phenomenon. Can it tell us anything about Lebesgue measure in high-dimensional spaces? Is there some analagous measure-theoretic property of high-dimensional spaces that 'almost' fails?