interior of a level set

Question

Let $F(z):\mathbb{C}\to\mathbb{R}$. Suppose $F(z)=l$ is a level set that we can show is a Jordan curve in the complex plane (or $R^2$). Under what condition on $F(z)$, we can say that $F(z)<l$ is located in the interior of $F(z)=l$ and $F(z)>l$ is located in the interior of $F(z)=l$?

Answer 1

the simple condition that does this is that the Hessian matrix of the function (of two real variables) is positive definite.

After that, one may allow that there is a global diffeomorphism that carries $F$ to a convex function.

Answer 2

I'll only consider the zero level set so "positive" and "negative" make sense. It's enough if $F$ is continuous, the interior and exterior of the zero level set are path connected sets, and there are two points $x_I$ in the interior and $x_O$ in the exterior where $F$ takes opposite sign. Then for any $x$ in the interior, there is a curve from $x$ to $x_I$ in the interior of the level set (i.e. $F$ is never zero), so by the intermediate value theorem, $x$ has the same sign as $x_I$. Likewise, any point outside the zero level set has the same sign as $x_O$.