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.

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 $B^{n}_1$.

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 $B^{n}_2$.

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 $p$?

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.

  • $\begingroup$ you are probably looking for: mathoverflow.net/questions/9185/… $\endgroup$ – Suvrit Feb 7 '12 at 21:16
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    $\begingroup$ so it seems that you are looking for uniform distribution on the surface of an $\ell_p$ ball (not in the ball). $\endgroup$ – Suvrit Feb 7 '12 at 22:10
  • $\begingroup$ I think you intend to normalize by $D^{1/p}$ instead of $D$, if I'm not mistaken. Also, $B_1^n$ is not the standard simplex. To generate uniformly on the $\ell_1$ ball you need to do something like multiply each coordinate $X_i$ by iid random variables $\epsilon_i$ uniform on {−1,+1}. $\endgroup$ – cardinal Feb 8 '12 at 3:22

The result you want, I think, is in Stationarity, Isotropy and Sphericity in $l_p^*$. It is behind a pay-wall, but the form of the distribution is stated in the abstract.

  • $\begingroup$ This paper fails to do the job since the OP wants independent $X_i$. $\endgroup$ – Mark Meckes Feb 8 '12 at 17:28
  • $\begingroup$ Sure, I misread the question as asking what form of iid distribution leads to particular forms of exchangeability via de Finetti representation theorems. This is the context I'm most used to seeing the normal distribution referred to a "spherically symmetric". The paper I linked to is the natural extension of that idea to the $l_p$ setting. $\endgroup$ – R Hahn Feb 8 '12 at 17:58

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 this paper by Barthe, Csörnyei, 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 $1 \le p < \infty$, which coincides with uniform measure only for $p=1,2$.


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