Hey Kreader12.

I understand your intuition and I think the best way would be to show that a line exists in the form ta + (1-t)b has points for t in [0,1] where it is outside the plane (but a and b are on the deformed plane).

One thing though I would consider is the Hessian matrix and use that to determine concavity. I think there are results that link the Hessian with concavity (or convexity) but you would need to look them up.