TITLE: "Paradoxes" in $\mathbb{R}^n$
QUESTION [24 upvotes]: 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 $n \to \infty$.

REPLY [13 votes]: 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):