## Thursday, December 18, 2008

### Back by popular demand

My anxious readership has been flooding me with emails, demanding to know if I'm still alive and why I quit blogging. (Just kidding. Is anyone still reading this thing?)

A good chunk of my time has been occupied by administrative activity (job search), as well as personal matters (both good and bad).

I regularly attend two courses: one by Gideon Schechtman and one by Itai Benjamini.

Here is a nice "paradox" from Gideon's first lecture. Consider the 2x2 square = [-1,1]^2. In each quadrant, inscribe a unit circle. Let S_2 be the largest circle about the origin not intersecting any of the inscribed unit circles. Let R_2 be its radius; compute R_2 (it's easy!).

Now repeat the same in n dimensions: divide [-1,1]^n into 2^n orthants, inscribe a unit ball in each one, and let R_n be the radius of the maximal ball about the origin that does not intersect any of the inscribed balls. It shouldn't take you more than 2 minutes to come up with a formula for R_n -- the basic 2-dimensional intuition carries over to higher dimensions.

However, to anyone not familiar with high-dimensional geometric phenomena, something very surprising should happen. I won't give it a away here, but feel free to discuss in the comments. This sort of "paradox" probably has a name -- anyone know what it is?