Let $\mathbb{S}$ be a closed and bounded convex body in 2-D with some non-empty intersection with positive quadrant and let it also contain origin. Let $c>0$ be the right-most point on the x-axis such that $(c,y)\in \mathbb{S}$ for some $y$. Define the function \begin{align} f(x)=\max_{(x,y)~\in~\mathbb{S}}y ~~,x\in[0,c] \end{align} Clearly,for a given $x'$, $f(x')$ is the northernmost point in the vertical strip $x=x'$, is $f(x)$ a concave function? (or does it have some nice properties.).

If you look at $f(x)$, it is the pointwise supremum of an affine function. And also, all the examples I can imagine is concave.