Spikey Spheres (2010)

Something I worked on in my PhD was analyzing high dimensional bodies via their "sections."

Here's the Busemann Petty Problem:

Given two origin symmetric convex bodies K and L in n dimensions. Suppose for every linear hyperplane A (passing through the origin) we have vol_{n-1}(K intersect A) \leq vol_{n-1}(L intersect A).

Is it true that vol_{n}(K) < vol_{n}(L)?

[Here vol_k should be thought of as length when k = 1, area when k = 2, and volume in the traditional sense in k = 3.... generalizes quite well to arbitrary dimensions. And sections are these quantities L (resp. K) intersect A]

Turns out the answer is NO! In n \geq 10, it can be explained with the simple examples of K and L being the unit volume (vol_n) cube and a euclidean ball of volume (vol_n) slightly less 1 respectively. Comes from Keith Ball who, in his PhD thesis, established that {n-1}-section volume of the unit volume cube lies in [1, \sqrt(2)]. However for the euclidean ball of unit volume the section volume is at least sqrt(2). So you can start with the unit volume ball, decrease its radius infinitesimally so (the n-1 section volume falls less than the n-volume does) and generate a clear counterexample.

What this looks like is a ball with volume less than a cube but section volume seemingly leaks out of the faces of the cube. So a "spikey ball," if you may.

A sphere of course have sum of the squares of all points equal to one. So in high dimensions, it’s clear that most points have to be close to 0. Hence the “spikes”

Boxes are more intuitive

I think it's becoming obvious we need better metaphors for high-dimensional spaces than "it's like geometry except not at all".

At the end of it all, we have a big list of numbers (a vector) where each position in the list (component/dimension) implies a specific "meaning" that we don't know. We also have a variety of well-known mathematical operations we can do on those lists, the effects of which may depend on the number of positions present (the dimensionality of the vector space).

The challenge would be to find a good intuitive model to explain those effects (and ideally a way to visualise the lists that preserves the effects). Saying "it's an 1800 dimensional sphere" satisfies neither of those properties: You cannot visualise it and even if you want to think about it theoretically, it has none of the intuitive properties of a 2D or 3D sphere.

This video (I know sorry) will help out with dimensionality in many computing problems, which isn't the concept here, or what many people think of infinite or high dimensionality.

It is worth your time IMHO.

https://youtu.be/q8gng_2gn70

I think the effect is almost even more pronounced if you go in the other direction and look at a 1D space, i.e. the real line.

Then both a sphere with radius 1 and a box with side length 2 will become the same interval, with no difference.

In the OP setup, the middle sphere would vanish completely and (modulo some annoyances at the interval boundaries) become either the empty set or a single-element set, in either case having both radius and volume of zero.

Is the sphere spikey or its shadow in lower dimensional space? Like how the shadow of a disk from the side would look like a long line.

I love the counter-intuition of high-dimensional spaces, seems to be making the rounds on my feeds these days.

One of the harder generalizations to develop intuition for is the fact that the measure of a d-sphere tends to 0 as d approaches infinity, even though for all d = 0, 1, 2, 3 that our meager brains can visualize, the opposite is true! Geometry goes crazy.