Convex range, peak to peak, or deviation function

Question

I have a non-negative function $f(a,x)\geq 0$ which is convex in its second argument, such that $$f(a,\lambda x_{1}+(1-\lambda)x_{2})\leq \lambda f(a,x_{1})+(1-\lambda)f(a,x_{2})$$ for every $a$ and $\lambda\in [0,1]$.

I am interested to study the behavior of the range of $f$ with respect to $a$, i.e., $$g(x)=\max_{a}f(a,x)-\min_{a}f(a,x).$$

1. Under which circumstances is $g(x)$ convex?
2. Are there other ways to measure the "deviation" in $f(a,x)$ w.r.t $a$, such that the result is zero if $f(a_{1},x)=f(a_{2},x)$ for every $a_{1},a_{2}$, and the deviation measure is a convex function?