Assume that we have a known multivariate distribution of $N$ variables denoted by $f(\vec{X})$.
Consider the random variable $Y = g(\vec{X})$ defined by a smooth function $g:\mathbb{R^{n}}\rightarrow\mathbb{R}$. I wish to find the function $g$ which maximize the entropy of $Y$ given $f(\vec{X})$.
Solutions to the general problem under any non-trivial constraints are of interest to me.
I tried using variational calculus for this task but couldn't come up with something other than the fact that $Y$ should be uniformly distributed, yet I still can't seem to find the form of $g$ given $f(\vec{X})$. I'm looking for any reference for a solution under any constraint on $g$, even in special cases(such as multivariate gaussian, or iid random variables).