Let $S_n = \{(p_1, \cdots, p_n)| p_i\geq 0, \sum_{i=1}^n p_i = 1\}$. I.e. $S_n$ is a $n-1$-simplex. Let $$ E(p_1, \cdots, p_n) = -\sum_{i=1}^n p_i\ln p_i $$ be the Shannon entropy.
I am stuck with the question, what is the $n-1$-Volume of $\{(p_1, \cdots, p_n)\in S_n \mid E(p_1, \cdots, p_n)>a\}$?
I mean I am interest in integral $$ \int_{X}dV $$ where $X= \{(p_1, \cdots, p_{n-1}) \mid p_i\geq 0, p_n = 1-\sum_{i=1}^{n-1}p_i\geq 0, E(p_1, \cdots, p_n)>a\}\subset {\mathbb R}^{n-1}$
Any non-trivial bounds are also interesting.
Just some values from numerical simulation. The horizontal axis is the entropy (in nats), and the vertical axis is the normalized volume.
The entropy tends to get concentrated around the mean, which is given by
$$ e(n)=\psi_0(n+1) + \gamma -1 \approx \log n -0.423 +\frac{1}{2n} $$
where $\psi_0$ is the digamma function.