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.