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.