I have a function of two variables, and I have checked that along one direction (fixing another variable), it is a monotonically increasing and concave function. Whereas in another direction (fixing the other), it is a concave function. Can I conclude that the joint function is concave?
1 Answer
First a bit of a degenerated example. Let $\ f:\mathbb R_+^2\rightarrow\mathbb R\ $ be given by:
$$ h(x\ y)\ :=\ x\cdot y $$
Along the argument straight lines parallel to one or the other axis the function is concave (since it's simply linear). But along the line $\ y=x\ $ the function is convex (it's a parabola).
Straight lines are concave in a degenerated way. It's possible to have strict concavity in place of straight lines; actually it's possible to have $\ \mathbb R^2,\ $ and the non-degenerated concavity+convexity.
I had domain $\ \mathbb R_+^2\ $ rather than $\ \mathbb R^2\ $ to make sure that the function is increasing on the lines parallel to the axes; actually, it's possible to have $\ \mathbb R^2,\ $ and non-degenerated concavity+convexity.