  We investigate positivity sets of nonnegative supersolutions of the fully
nonlinear elliptic equations $F(x,u,Du,D^2u)=0$ in $\Omega$, where $\Omega$ is
an open subset of ${\mathbb R}^N$, and the validity of the strong maximum
principle for $F(x,u,Du,D^2u)=f$ in $\Omega$, with $f\in\text{C}(\Omega)$ being
nonpositive. We obtain geometric characterizations of positivity sets
$\left\{x\in\Omega\,:\, u(x)>0\right\}$ of nonnegative supersolutions $u$ and
establish the strong maximum principle under some geometric assumption on the
set $\left\{x\in\Omega\,:\, f(x)=0\right\}$.
