Convex sublevels of non-convex function

Question

For a function to be quasi convex, the requirements are:

1. The domain is convex.
2. The sublevel sets are convex.

In my understanding, this function is not convex as it has a negative second derivative.

In the figure, the first condition is satisfied. Regarding the second condition, if we see the sublevel for $\beta$ i.e. $ S_\beta $ where $f(x)<\beta$, the section where the function has negative second derivative gets included. So why is the sublevel $ S_\beta $ considered as convex ?

Answer 1

It's not convex it's quasiconvex. A quasiconvex function means that the inverse image of any set of the form $(-\infty,a)$ is convex. Graphically this means the inverse image of all the points below $\beta$ must be convex and since it is the interval $(-\infty,c)$ the function is quasiconvex.