High-dimensional phenomena in geometry

 Our geometric intuition, built on lower dimensional experience, can often be misleading when thinking about high-dimensional problems. This isn't anyone's fault; I think reasoning about/visualising is genuinely difficult. It should therefore not be surprising that there are many surprises in high dimensions, but surprising they still are at first glance. These include exotic spheres, where the pioneering work was done by Milnor and then Kervaire,  the existence of compact convex sets whose set of extreme points is not closed in dimension 3 or more (but impossible in dimension 2), positive polynomials not reaching its infimum, the wild west that is 4-manifold topology, and Borsuk's conjecture (currently known to be false in dimensions at least 64) to name but a few.  So many of these would merit their own (series of) blog posts, but here are the ones related to geometry which I thought are worth sharing. 

We begin with the volume of a ball $B^d$ in Euclidean space $\mathbb{R}^d$. Some amount of integral calculus shows that the volume of an $n$-ball of radius $R$ satisfies the following recursion:

\begin{equation}
V_d(R) =
    \begin{cases}
    1 & d=0 \\
    2R & d=1 \\
    \frac {2\pi }{d}R^{2}\times V_{d-2}(R) & {\text{otherwise}}
    \end{cases}
\end{equation}
The closed form expression for this is $\frac {\pi ^{d/2}}{\Gamma {\bigl (}{\tfrac {d}{2}}+1{\bigr )}}R^{d}$. By applying Stirling's approximation, for any fixed $R$ the volume of an $R$-ball tends to 0 as $d$ goes to infinity. One can also show that “most” of the volume of the $d$-dimensional sphere is contained near the boundary of the sphere. More specifically, for a $d$-dimensional sphere of radius $r$, most of the volume is contained in an annulus of width proportional to $\frac{r}{d}$ (all of the above so far are good exercises for secondary school students). 

As a corollary, the following should be quite believable: most of the mass of a unit cube is near the corners. This is because the biggest inscribed sphere in the middle has vanishing mass. 

Let now $A_d(R)$ denote the volume of a $d$-dimensional sphere embedded in $d+1$-dimensional Euclidean space. $A_{d-1}(R)={\frac {dV_{d}(R)}{dR}}$ since a ball is a union of concentric spheres and increasing the radius by $\epsilon$ sweeps out a shell of thickness $\epsilon$. Thus, $V_{n}(R)={\frac {R}{n}}A_{n-1}(R)$.
Consider the following map between the unit sphere $S^{d+1}$ and $S^1 \times B^d$:
\begin{equation}
    (x,y,{\vec {z}})\mapsto \left({\frac {x}{\sqrt {x^{2}+y^{2}}}},{\frac {y}{\sqrt {x^{2}+y^{2}}}},{\vec {z}}\right)
\end{equation}
where $|(x,y,{\vec {z}})|=1$; this is a bijection up to ignoring sets of measure 0. Volume is preserved because at each point; the difference from isometry is a stretching in the $xy$ plane (by $1/\!\sqrt {x^{2}+y^{2}}$ times in the direction of constant $x^{2}+y^{2}$) that exactly matches the compression in the direction of the gradient of $\vec{z}$. This implies that $A_{d+1}(R)=(2\pi R)V_{d}(R)$, and hence that the volume of the $d$-sphere also tends to 0. 

There is one big surprise lurking here. Suppose one takes a small slice around the equator of the 2-sphere. Then the surface area is quite small. However, a calculation shows that as the dimension $d$ tends to infinity, a small slice around the equator contains asymptotically \emph{all} the volume of the sphere, and this doesn't depend on the equator! One can use this for estimating integrals for instance: if the dimension is high enough, just pick an equator and do the integral over that.

Spheres and cubes

In two dimensions, the unit square is completely contained in the unit sphere. The distance from the centre to a vertex (radius of the circumscribed sphere or length of the diagonal of the cube) is $\frac{\sqrt2}{2}$ and the apothem (radius of the inscribed sphere) is $\frac{1}{2}$. In four dimensions, the distance from the centre to a vertex (i.e., the length of the cross-diagonal) is 1, so the vertices of the cube touch the surface of the sphere. However, the length of the apothem is still $\frac{1}{2}$. The result, when projected in two dimensions, no longer appears convex; however all hypercubes are convex. This is part of the strangeness of higher dimensions - hypercubes are both convex and “pointy.” In some directions it has diameter 2, but in others arbitrarily large diameter as $d$ goes to infinity. In dimensions greater than 4 the distance from the centre to a vertex is $\frac{\sqrtd }{2} > 1$, and thus the vertices of the hypercube extend far outside the sphere.

A related phenomenon is the following, which I learned from Remi Coulon's website.

Take a $d$-dimensional hypercube of side length 1, subdivide it into cubes of side length $\frac{1}{2}$, and add an inscribed ball in each of these cubes. One can add a ball of radius $r_d= \frac{\sqrt{d}}{4}- \frac{1}{4}$ between all the inscribed balls, which in particular doesn't fit in the unit cube! Again, this is due to the rather strange shape of the hypercube. Visualisations can be found on Coulon's website.

Most of what I said is well-known and exists in many different shapes and forms in various online and print sources. I should credit these lecture notes as my main reference.

Busemann-Petty problem

The Busemann-Petty problem asks the following: if $K, T$ are symmetric convex bodies in $\mathbb{R}^n$ such that $\mathrm {Vol} _{n-1}\,(K\cap A)\leq \mathrm {Vol} _{n-1}\,(T\cap A)$ for every hyperplane $A$ passing through the origin, is it true that $\mathrm {Vol} _{n}\,(K)\leq \mathrm {Vol} _{n}\,(T)$? Busemann and Petty showed the answer is yes when $K$ is a ball. I learned about its history from this excellent MO answer, which I partially quote.

Many mathematician's gut reaction to the question is that the answer must be yes and Minkowski's uniqueness theorem provides some mathematical justification for such a belief---Minkwoski's uniqueness theorem implies that an origin-symmetric star body in $\mathbb{R}^n$ is completely determined by the volumes of its central hyperplane sections, so these volumes of central hyperplane sections \emph{do} contain a vast amount of information about the bodies. It was widely believed that the answer must be yes. 

However, counterexamples began to arise in many dimensions. The easiest one to understand, that also fits today's theme, is this: in 1986, Keith Ball proved that the maximum hyperplane section of the unit cube is $\sqrt{2}$ regardless of the dimension, and a consequence of this is that the centred unit cube and a centred ball of suitable radius provide a counter-example when $n \ge 10$. Subsequently, counterexamples in all dimensions at least 5 were given.

It was shown that the Busemann-Petty problem has an affirmative answer in $\mathbb{R}^n$ iff every origin-symmetric convex body in $\mathbb{R}^n$ is an intersection body. But the question of whether a body is an intersection body is closely related to the positivity of the inverse spherical Radon transform. In 1994, Richard Gardner used geometric methods to invert the spherical Radon transform in three dimensions in such a way to prove that the problem has an affirmative answer in three dimensions. 

In general, the answer is yes in dimensions at most 4 and no in higher dimensions. The proof of this involves a certain amount of getting one's hands dirty, and Zhang Gaoyong, who eventually settled the problem, does so in a relatively clean way. However, I still don't want to present that proof. What's arguably even more interesting is the history: Zhang originally claimed that the answer is negative in dimension 4, but his claimed counterexample was refuted by Koldobsky, who provided a new Fourier analytic approach to convex bodies and established a very convenient Fourier analytic characterization of intersection bodies. After this, Zhang quickly proved that in fact every origin-symmetric convex body in $\mathbb{R}^4$ is an intersection body and hence that the Busemann-Petty problem has an affirmative answer in $\mathbb{R}^4$---the opposite of what he had previously claimed. Both papers were published in the Annals, which makes the set of results published there inconsistent!