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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

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  • $\begingroup$ 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.) $\endgroup$
    – p.s.
    Commented Oct 27, 2015 at 22:54
  • $\begingroup$ 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? $\endgroup$
    – majmun
    Commented Oct 28, 2015 at 15:28

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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 $a$.


Caveat: the definition you have in mind is the one prevalent in optimization and assorted applications of mathematics. In calculus of variations, there is another property that goes by the same name; occasionally qualified as Morrey quasiconvexity.

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