Is $f(x,y)=y\left(2^{\frac{x}{y}}-1\right)$ a strictly convex function when $x\ge0,y\ge0$?

Question

Is function $$f(x,y) = y\left(2^{\frac{x}{y}}-1\right)$$ strictly convex when $x\ge0,y\ge0$?

I can show its Hessian matrix is positive semidefinite, but it is only a sufficient condition for the strictly convex. Any help is appreciated.

Robert's right below, of course. However: when you computed the Hessian, were you able to see/prove that it wasn't positive definite? That is, were you able to find a direction $v$ such that $Hv=0$? If so, then you're done—it's not strictly convex. — Michael Grant, Jun 28 '18 at 15:16

score 2 · Accepted Answer · answered Jun 28 '18 at 03:49

2

Hint: Consider the line $y=cx$ for constant $c$.

answered Jun 28 '18 at 03:49

Robert Israel

1 Answers1