  We prove that if a level set of a degree $n$ general inverse $\sigma_k$
equation $f(\lambda_1, \cdots, \lambda_n) = \lambda_1 \cdots \lambda_n -
\sum_{k = 0}^{n-1} c_k \sigma_k(\lambda) = 0$ is contained in $q + \Gamma_n$
for some $q \in \mathbb{R}^n$, where $c_k$ are real numbers not necessary to be
non-negative and $\Gamma_n$ is the positive orthant, then this level set is
convex. As an application, this result justifies the convexity of the level set
of all general inverse $\sigma_k$ type equations, for example, the
Monge--Amp\`ere equation, the Hessian equation, the J-equation, the deformed
Hermitian--Yang--Mills equation, the special Lagrangian equation, etc.
Moreover, we find a numerical condition to verify whether a level set of a
general inverse $\sigma_k$ equation is contained in $q + \Gamma_n$ for some $q
\in \mathbb{R}^n$, which is a way to determine the convexity of this level set.
