I would like to attempt to solve the given problem below. I am sorry that I dont know whether such a question is solvable at all, even for just one specific example. If the solution set is empty, I am also helpless about how to show it. I asked this question two years ago at math.stackexchange with no answer or helpful comment.
Given two distinct and continuous probability density functions on real numbers, $f_0$ and $f_1$ consider the following set of density functions:
$$\mathscr{G}_0=\left\{g_0:\int_{\mathbb{R}}\log\left(\frac{g_0(y)}{f_0(y)}\right)g_0(y)\mathrm{d}y\leq \epsilon_0\right\} $$ and $$\mathscr{G}_1=\left\{g_1:\int_{\mathbb{R}}\log\left(\frac{g_1(y)}{f_1(y)}\right)g_1(y)\mathrm{d}y\leq \epsilon_1\right\} $$
for some sufficiently small $\epsilon_0$ and $\epsilon_1$ such that $\mathscr{G}_0$ and $\mathscr{G}_1$ are also distinct. In other words any density $g_0\in \mathscr{G}_0$ is not an element of $\mathscr{G}_1$ and vice verse. Note that $f_0$ and $f_1$ are known and assumed to be positive so that the terms $\log(g_i/f_i)$, $i\in\{0,1\}$, are well defined.
Question: Are there a pair of density functions $g^*_0\in \mathscr{G}_0$ and $g_1^*\in \mathscr{G}_1$ such that
$$g^*_0=\arg\sup_{g_0\in \mathscr{G}_0}\left(\inf_{u\in \mathbb{R}}\ln \int_{\mathbb{R}}\left(\frac{g_1^*(y)}{g_0^*(y)}\right)^u g_0(y)\mathrm{d}y\right)$$ and $$g^*_1=\arg\sup_{g_1\in \mathscr{G}_1}\left(\inf_{u\in \mathbb{R}}\ln \int_{\mathbb{R}}\left(\frac{g_1^*(y)}{g_0^*(y)}\right)^u g_1(y)\mathrm{d}y\right)?$$
