I have two variables, $x$ and $y$, and a few inequalities of the form $f(x,y) \le g(x,y)$.
I want to know if the intersection of all $(x,y)$ that satisfy each inequality is convex. Is there some generic way to do it? Maybe based on second order derivative (or the Hessian in this case), similarly to the test whether a function is convex?
Finding whether one inequality defines convex set is also good, because if they all define convex sets, then their intersection must define a convex set as well.
Thanks.
Best Answer
Take $h(x,y)=f(x,y)-g(x,y)$ and if this function $h$ is convex then $h(x,y)\leq0$ is a convex set in $\mathbb{R}^2$. However in general there could be functions $f,g$ which are not convex, but where $f(x,y)\leq g(x,y)$ is still convex (see also other answers). If there is a general method then it should be beyond just convexity of $f-g$.