"Paradoxes" in $\mathbb{R}^n$

Question

One may think of this question as a duplicate of this one. I see it more like an extension.

The "inscribed sphere paradox" discussed in the aforementioned question states that if you inscribe a sphere in an $n$-dimensional cube of side $1$, then the volume of the sphere goes to $0$ as $n \to \infty$, while the volume of the cube remains the same.

A more involving paradox is Hamming's Four Circle paradox, also described as an answer to that post.

A more straightforward paradox (also discussed earlier by R. C. Hamming) is the fact that the angle $\theta$ between the diagonal of a cube $(1,1,\ldots,1)$ and any direction $(0,\ldots,1,0,\ldots,0)$ satisfies $$\cos \theta = \frac{1}{\sqrt{n}} \to 0 \mbox{ as } n \to \infty.$$ This means that, as $n$ increases, the diagonal is almost perpendicular to all $(0,\ldots,1,0,\ldots,0)$ (almost lying in (all!) corresponding hyperplanes).

My question is: are there any other elementary examples of these so-called "paradoxes" (for instance, for other objects than sphere/cubes)? I am thinking more of elementary examples in which the intuition from simple plane geometry ($\mathbb{R}^2$) fails miserably in $\mathbb{R}^n$, particularly when $n \to \infty$.

Answer 1

The unit ball in ${\bf R}^n$ has a lot of counterintuitive properties as $n \to \infty$. Almost all of the volume of the ball is concentrated near the boundary. In fact almost all the volume is concentrated near the equator. That is, for large $n$ if you choose a point in the unit ball at random it is likely to nearly lie on the unit sphere and to have first coordinate approximately zero.

The size of an orthonormal basis in ${\bf R}^n$ is exactly $n$. However, the maximal size of a set of unit vectors $v_i$ for which $|\langle v_i, v_j\rangle| < 10^{-10}$ for all $i \neq j$ grows exponentially as $n \to \infty$. (A simple volume computation shows this.)

In my second answer here I pointed out that in ${\bf R}^n$ for large $n$ we can have convex sets $A \subset B$ such that $B$ is contained in the $\epsilon$-neighborhood of $A$ but the centroid of $B$ is far away from the centroid of $A$.

Answer 2

For the $n$-simplex $$\Delta_n = \{x\in\mathbb{R}^{n+1}\ :\ x_i\geq 0, \sum x_i = 1\},$$ its "midpoint" $m = [1,\dots, 1]/(n+1)$, and its corners $e_k$ it holds that $$\|m - e_k\|\to \infty$$ while the distance of the midpoint to the $n-1$ dimensional faces goes to zero (both for $n\to\infty$).

I found it quite counterintuitive that for a convex body like the simplex it can happen that its corners move apart from the midpoint while its "sides" move closer to it. But actually, this is somehow that standard picture of a high dimensional convex body as I learned from these slides by Roman Vershynin (who attributes this finding to V. Milman):

Answer 3

Given any aperture less then π, when the dimension of space is large enough, a one-sided cone with such aperture can be fitted in an orthant (hyperoctant).}

Actually, it is the contrary. The widest circular cone that can be fitted into an orthant (hyperoctant) has the aperture $$ \varphi=\arccos\sqrt{\frac{n-1}{n}}$$ which approaches zero as n grows to infinity.

This is connected with the other responces that describe how faces of the cube come closer to the middle, but is counterintuitive as one would expect that "there is more room" in higher dimensions. Maybe, it is just that the circular cones capture more of the space, so there is less leftover?

This could be judged as selfpromotion, as it is the closing remark of my article, but it is relevant to the question. If there are other sources to the fact or related articles, I would feel grateful for any comments.