Sampling uniformly from a sphere - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T01:27:04Z http://mathoverflow.net/feeds/question/87827 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/87827/sampling-uniformly-from-a-sphere Sampling uniformly from a sphere Erik Aas 2012-02-07T18:57:55Z 2012-02-08T17:31:43Z <p>Let $B^{n} _p=${$(x_1, \dots, x_n) : |x_1|^p + \dots |x_n|^p = 1$} be the unit ball in $\mathbb{R}^n$ in the $\ell^p$ norm.</p> <p>If $X_1,\dots,X_n$ are iid $\exp(1)$ -distributed random variables, then $(X_1/D,\dots,X_n/D)$, where $D =X_1+ \dots + X_n$ is uniformly distributed in $B^{n}_1$.</p> <p>If $X_1,\dots,X_n$ are iid normally distributed with mean 0, then $(X_1/D,\dots,X_n/D)$, where $D = (X_1^2+\dots+X_n^2)^{1/2}$, is uniformly distributed in $B^{n}_2$.</p> <p>Is there a choice of $X_1,\dots , X_n$ iid such that $( X_1 / D, \dots, X_n/D)$, where $D = (|X_1|^p + \dots + |X_n|^p)^{1/p}$ is uniformly distributed in $B^{n} _p$ for arbitrary $p$?</p> <p>I would be happy with any sensible common generalization of the two statements above. I have no particular reason to believe there is such a generalization - I'm just hoping that two so similar and neat examples have similarly nice generalizations.</p> http://mathoverflow.net/questions/87827/sampling-uniformly-from-a-sphere/87843#87843 Answer by R Hahn for Sampling uniformly from a sphere R Hahn 2012-02-07T21:34:18Z 2012-02-07T21:34:18Z <p>The result you want, I think, is in <a href="http://www.springerlink.com/content/r2771gx9j2g40132/" rel="nofollow"> Stationarity, Isotropy and Sphericity in $l_p^*$</a>. It is behind a pay-wall, but the form of the distribution is stated in the abstract.</p> http://mathoverflow.net/questions/87827/sampling-uniformly-from-a-sphere/87907#87907 Answer by Mark Meckes for Sampling uniformly from a sphere Mark Meckes 2012-02-08T17:31:43Z 2012-02-08T17:31:43Z <p>If by uniform measure you mean $(n-1)$-dimensional Hausdorff measure on the sphere, the answer is no. As a consequence of the results of <a href="http://www.cims.nyu.edu/~naor/homepage%20files/product.pdf" rel="nofollow">this paper</a> by Barthe, Cs&ouml;rnyei, and Naor, under mild regularity assumptions the only measure on the boundary of any convex body which can be generated in this way is the "cone measure" on the $\ell_p$ sphere for $1 \le p &lt; \infty$, which coincides with uniform measure only for $p=1,2$.</p>