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?


Anonymous said...

Yes, your blog made it into my google reader subscriptions a while ago. I'd actually love to see some coverage of Schechtman's course that you pointed out. The first part of the lecture notes looks pretty interesting and if you find the time it would be great to read some summaries and highlights of this course on your blog.

Anonymous said...

Btw, the exercise you described also appears in Steele's Cauchy-Schwarz Master Class.

Aryeh said...

Really -- where? I have that book, and ever read it at one point...

Jerm Pollet said...
This comment has been removed by a blog administrator.