Given the entropy fucntion:
$$ H = -\sum_i^n p(i) \ln(p(i))\,.$$
where $p(i)$ are probabilities and $n=4$, I would like to know all the points in the probability space that make $H = k$, being $k$ a given constant. I've done some numeric aproximations, but I would like to know if the solution can be expressed as an elementary function. If it doesn't exist, where could I find the proof?
Thanks in advance