Given two densities $f$ and $g$ the squared Hellinger distance between them is defined as follows
$ S(g,f)=H^2(g,f)=\frac{1}{2}\int_{\mathbb{R}}\left(\sqrt{g(y)}-\sqrt{f(y)}\right)^2 \mbox{d}y $
I build a set
$ {\cal{F}}=\{g:S(g,f)\leq \epsilon\} $
with respect to the squared Hellinger distance and check if it is compact.
Question: Is this set ${\cal{F}}$ compact?
What I know:
$1$- ${\cal{F}}$ should be a convex set because the function that creates this set, $\left(\sqrt{g(y)}-\sqrt{f(y)}\right)^2$, is convex.
$2$- Hellinger distance satisfies the first three axioms of a metric http://en.wikipedia.org/wiki/Metric_%28mathematics%29 but not the triangle inequality (although I dont have any counter example, I read it in a paper which states it shortly in pharenthesis)
$3$- If ${\cal{F}}$ was a metric, I could say that it was compact because all continuous real functions of this set has a maxima.
Thanks in advance for reading this post.