Is it a jointly concave function of two variables?

Question

I have following function $f(x,y)=\min(y(1-x),xya)$ where $x\geq0,y\geq0,a\geq0$. Only $x$ and $y$ are the optimization variables. I want to ask is it a concave function of $(x,y)$? Any help in this regard will be much appreciated.

Answer 1

No it is not

if $a > 0$: On the line $y =x$ function is $f(x,y)=\min(- x^2 + x,ax^2) $ so $f$ agrees with $a x^2$ near zero i.e., $f(x) = ax^2$ on small interval $[0 , \epsilon)$. which is not concave. Also it is clear that $f$ is not convex too.

EDIT

If $a = 0$ then on the segment $\{ (1+t , 1-t) ~ : ~ 0 \leq t \leq1\} $

$f$ agrees with $t^2 -t$ which is not concave, clearly $f$ it is no convex too.