I need your expertise in understanding the following:
Let $n \in \mathbb{N}$, $x_i \in \mathbb{R}$ for every $i \in [n]$ and let $a \in \mathbb{R}_+$.
What can be said about the following in term of convexity (let $j$ be any arbitrary integer such that $i \in [n]$:
$$ \frac{\frac{a^2}{2n} + \max\left\lbrace 0, 1 - x_i\right\rbrace} {\frac{a^2}{2} + \sum\limits_{j \in [n]} \max\left\lbrace 0, 1 - x_j\right\rbrace} $$
I am asking this since, it's easy to see that both the denominator and nominator are convex (it resembles the objective function of SVM), however is this fraction convex, or quasi-convex, concave, etc... ?
Please advise and thanks in advance.
P.s. A more advanced question would be, what can be said on the fraction of two convex function in general?
For simplicity, consider the case where $f$ and $g$ are convex, twice differentiable functions on an interval and $g > 0$. We have
$$ \left(\frac{f}{g}\right)'' = \frac{f'' g^2 - 2 f' g g' - f g g'' + 2 f (g')^2}{g^3} $$
and the condition for $f/g$ to be convex is that the numerator is always nonnegative. Unfortunately, not a very nice condition!