What is this exponential distribution called?

Question

The symmetric Dirichlet distribution is given by:

$P(X)=Z\prod{{x_i}^{\lambda-1}}$

where $Z$ is a normalization factor, $x_i\geq0$, and $\sum x_i=1$.

I am wondering about the following related distribution, where we replace $x_i$ with $e^{x_i}$:

$P(X)=Z\prod{e^{x_i(\lambda-1)}}$

where again $Z$ is a normalization factor, $x_i\geq0$, and $\sum x_i=1$.

Does the following distribution have a name? What is known about it (does it occur naturally somewhere)?

Answer 1

$P(X)=Z\prod e^{x_i(\lambda-1)}=Ze^{(\lambda-1)\sum x_i}=Ze^{\lambda-1}$, so this distribution is constant, and therefore uniform, over the region $\{X:x_i\ge 0,\sum_i x_i=1\}$.