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