According to exercise 3.32 in Boyd & Vandenberghe's Convex Optimization, if both $f$ and $g$ are convex, positive and non-increasing (or non-decreasing) then $fg$ is convex. However, if we let $f(x,y)=x$ and $g(x,y)=y$ then over the non-negative orthant the conditions are fulfilled but the sublevel set for $0.5$ is not convex (which I think should be convex if the function is convex).
Where am I wrong in understanding the meaning of the result presented in this exercise?
Best Answer
To summarize the comments:
The theorem about the product of two positive non-decreasing convex functions being convex applies only to functions of one real variable.
For example, we can use it to conclude that $x e^x$ is convex for $x\ge 0$. We cannot use it to conclude that $xe^y$ is convex over $x, y\ge 0$.