My problem is as follows: Let f(x,y) be a general concave function (i.e. any concave function), and let S be points (x,y) such that f(x,y) greater than or equal to zero.
I understand that this means S is the set of points on or below f(x,y) and >0, and I can show how it is convex in a sketch/graph. I just don't understand how you show it mathematically in a sufficient way using the definition of a convex set.
Any help would be greatly appreciated.
If $\xi_0=(x_0,y_0)$ and $\xi_1=(x_1,y_1)$ are two points in $S$ then we must have $f(\xi_0)\geq 0$ and $f(\xi_1)\geq 0$. Let $0\leq t\leq 1$ and consider $\xi_t=(1-t) \xi_0+t\xi_1$. By concavity of $f$ we must have: $$ f(\xi_t) \geq (1-t) f(\xi_0) + t f(\xi_1) \geq 0$$ so also $\xi_t\in S$.