# High-dimensional geometry: Top-down Vs. Bottom-up

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?

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I think the answer is, strangely, exactly the same as the spiky ball thing. In high-dimensional spaces, almost all the measure of the unit ball is contained in the shell with $r>1-\epsilon$, thus convex bodies appear spiky and nonconvex. In infinite-dimensional spaces, you replace "almost all" with "all", and derive a contradiction - none of the measure is in the insides, so there is no measure anywhere. – Will Sawin May 28 '12 at 22:33
That's interesting, but I'm not quite convinced - I can't see why translation invariance should fail from this example. – Simon Lyons May 29 '12 at 9:30
@Simon: recall that translation invariance characterizes Lebesgue measure up to a constant; so a translation invariant measure on an infinite dimensional Banach space should put zero measure on any ball (since it would otherwise put infinite measure on any ball doubled - that's any ball). – Benoît Kloeckner May 29 '12 at 12:34
It's not really a mathematical answer, but the use of infinite dimensional techniques for high dimensional problems is quite common in physics (esp. the thermodynamic limit and continuum mechanics). My knowledge in this area is limited, but I'm sure there must be corresponding mathematical statements. – Ollie Margetts May 29 '12 at 15:52
For instance, the existence of phase transitions for large numbers of particles can be inferred from the behaviour of an infinite dimensional system. The partition function for the finite system is analytic, whereas for the infinite dimensional system it has singularities. The singularities correspond to phase transitions. – Ollie Margetts May 29 '12 at 15:56