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I'm reading The Elements of Statistical Learning. I have a question about the curse of dimensionality.

In section 2.5, p.22:

Consider $N$ data points uniformly distributed in a $p$-dimensional unit ball centered at the origin. suppose we consider a nearest-neighbor estimate at the origin. The median distance from the origin to the closest data point is given by the expression:

$$d(p,N) = \left(1-\frac{1}{2^{1/N}} \right)^{1/p}.$$

For $N=500$, $p=10$, $d(p,N)\approx0.52$, more than halfway to the boundary. Hence most data points are closer to the boundary of the sample space than to any other data point.

I accept the equation. My question is, how we deduce this conclusion?

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    When you cite a book, please quote _both_ its author and the title. (Are there really no upper-case letters in the title?)2011-12-11
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    Can be downloaded for free from http://www-stat.stanford.edu/~tibs/ElemStatLearn/download.html and took a while, it is 763 pages. Title and authors only at http://www.springer.com/statistics/statistical+theory+and+methods/book/978-0-387-84857-0?changeHeader2011-12-11

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