# Can something that is not a level curve be quasi-convex/concave?

Something is quasi convex (concave) if the lower (upper) contour is convex. I don't know if we can talk about an upper/lower contour for a function, not a function's level curves/sets. Therefore, can we talk about a function be quasi-concave/convex, or must we be discussing level curves/sets?

For example, think of $y = \frac{1}{x}$ over domain $[0,\infty]$. To me it seems like this satisfies $f(\lambda x + (1-\lambda)y \leq \lambda f(x) + (1-\lambda)y$, where $f(\cdot) = \frac{1}{\cdot}$ (perhaps this is my problem; maybe I am getting 2 and 3+ dimensional properties confused?) which would mean it is convex, but it also satisfies $f(\lambda x + (1-\lambda)y \geq \min \{f(x),f(y)\}$, and is thus quasi-concave. Therefore this would be convex, which implies quasi-convex, and also quasi-concave, meaning it is quasi-linear...

I have to be making some mistake here; I am just wondering if someone can point it out to me

• I don't see any mistakes in your reasoning. One dimensional monotonic functions are quasilinear (their sublevel and superlevel sets are intervals, which are convex.)
– p.s.
Commented Oct 27, 2015 at 22:54
• Their sub level and super level sets? I guess that is where my confusion is. wikipedia seems to say that the upper/lower contour sets are convex. And the contour sets wiki seems to discusses values of f(x) (stuff in $\mathbb{R}^2$), not the $x$s, the set of which would be intervals... Do you have any reference that says that we would look at the sub level sets? I mean, I guess looking at upper/lower contour in this case probably doesn't make much sense? Commented Oct 28, 2015 at 15:28

Yes, it makes perfect set to talk about quasiconvex and quasiconcave functions. Case in point: Wikipedia article Quasiconvex function. As you supposed, $f$ is quasiconvex if $\{ x:f(x)< a \}$ is convex for every $a$.