TITLE: Looking for Cover's hubris-busting ${\mathbb R}^{N\gg3}$ counterexamples
QUESTION [18 upvotes]: In lecture 4 of his Introduction to Dynamical Linear Systems course, right after interpreting the inner product in ${\mathbb R}^N$ in terms of the cosine of the subtended angle, Stanford's Stephen Boyd goes into humorous aside in which he pokes fun at the notion that one can "visualize" and reason about ${\mathbb R}^N$—for some $N$ greater than, say, $5$—by relying on intuitive analogies with ${\mathbb R}^2$ or ${\mathbb R}^3$.
(FWIW, I found Boyd's commentary pretty funny.  It starts at 59'46" of the YouTube video, and goes for about 3 minutes.)
At some point during this riff, Boyd remarks that "[the late Stanford professor Thomas] Cover has a series of examples showing that [when it comes to ${\mathbb R}^{N\gg3}$] you know nothing!"  He then adds something like "I should really collect some of these from him, 'cuz they're really good...  There are about four of 'em..."
I've been positively haunted by this mythical list of counterexamples ever since I heard Boyd's mention of it.

If anyone can point me to some version of Cover's list I'd greatly appreciate it.

The matter goes beyond satisfying my idle curiosity.  I feel this is an issue that really does need a bit of awareness-raising in the research community.  I'm surrounded by researchers who blithely rely on their ${\mathbb R}^3$-based intuitions to reason about algorithms that search through an "energy landscape" (i.e. a hypersurface) in ${\mathbb R}^{1\; \mathrm{bazillion}}$, and they poo-pooh any concern over the applicability of said intuitions, and the soundness of the reasoning based on them.  (Heck, even assuming that what holds in, say, ${\mathbb R}^2$ will necessarily hold in ${\mathbb R}^3$, or that what holds in ${\mathbb R}^1$ will necessarily hold in ${\mathbb R}^2$, can get one all wet...)
So my hope is that Cover's list will rattle them enough to show them their hubris.

REPLY [7 votes]: I also don't know about the list, but this simple geometric construction might be there. I remember it being attributed to Vitali Milman.

Divide $n$-dimensional cube $Q$ into $2^n$ equal cubes in the natural way;
Inscribe a ball in each of these $2^n$ cubes;
There is some room left at the center of $Q$; inscribe a ball there. 

This is how it works in two dimensions: $Q$ is the black square. 

You can imagine a similar $3$-dimensional picture with $8$ blue balls and a small red one in the middle. 
The red ball is contained in $Q$ when $n\le 9$, but not when $n\ge 10$.