Let $C\subset \mathbb{R}^n$ be a compact, convex set. In any convex analysis course, it would be a standard homework exercise to prove that the functions $f(x)=\max_{y\in C} \|x-y\|$ and $f(x)=\min_{y\in C} \|x-y\|$ are convex (although the proof of the latter is a little harder than that of the former, for all proofs I've seen). I'm interested in a generalization of these functions: say that $C$ has volume $1$, define $a\in(0,1)$, and define $$f_a(x)=\min\{r:\mathrm{vol}(B_r(x)\cap C)=a \}$$ where $B_r(x)$ is the ball of radius $r$ about $x$. The two functions I described above would correspond to the limiting cases where $a=1$ and $a\to0$ respectively. (I'm not sure if such functions have a name, like "partial distance", "threshold distance", or something to that effect)

I was able to convince myself that (for fixed $a$), the function $f_a(x)$ is convex, although I had to appeal to Minkowski sums and treat $f_a(x)$ as a jointly convex function $f_a(x,S)$ over the set of all measurable subsets of $C$. This seems like a really complicated way to prove this fact; is there something more direct that I was missing?