# Quasi concavity and Quasi Convexity-intuitive understanding

I'm having trouble grasping the concept of quasi concavity and quasi convexity.

My textbook states that if f is quasi-concave, then f (λx + (1 − λ) y) ≥ min {f(x), f(y)} . Also that is f is quasi convex, then f (λx + (1 − λ) y) $\leq$ max {f(x), f(y)}.

So that implies a function like $f(x) = x^2$ is quasi concave because it satisfies the definition of quasi concavity. But that function is also convex, hence it is also quasi convex?

How do you intuitively understand whether a function is quasi concave or quasi convex?

• $f(x)=x^2$ is not quasi-concave: take $x=-1$, $y=1$, $\lambda=1/2$.
– daw
Commented Jun 15, 2015 at 11:29
• It's not quasi concave. Check again. Commented Jun 15, 2015 at 11:31
• Also, that's not the definition of quasi convex that I learned 40 years ago. Maybe it was changed -- I haven't been paying attention for the last few decades. Commented Jun 15, 2015 at 11:32
• Is the definition wrong? I just double checked it- I got that off a video I watched, I have been trying to understand this from multiple sources and this was the one I understood most clearly. Commented Jun 15, 2015 at 11:37
• youtube.com/watch?v=5WmpKjnlFYE this one, btw Commented Jun 15, 2015 at 11:37

Consider the level sets of function $f$, $$N(f,a) = \{ x: \ f(x)\le a\}.$$ If $f$ is quasi-convex then the level sets $N(f,a)$ are convex for all $a$.
To see this, assume $f$ to be quasi-convex, $x,y\in N(f,a)$ for some $a$. Then all convex combinations of $x,y$ are in $N(f,a)$: $$f(\lambda x + (1-\lambda)y) \le \max(f(x),f(y))\le a \quad \forall \lambda\in(0,1).$$
• Okay, I have a little trouble understanding this so can you please tell me if my working is correct? Lets take the function $f(x) = x^2$. 1. Let a = 4. We get this bowl shaped curve below the straight line at y=4. 2. Now in that bowl, if we join any two points, those would lie above the bowl and hence would be less than the max of the function (i.e they'd lie below a) 3. Hence this is quasi convex. Commented Jun 15, 2015 at 11:47
• Thats ok. the important thing is to observe if you take two points, join them, then the joining line is below the line $y=a$.