On the Convexity of General Inverse ${\sigma }_k$ Equations

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 $q+{\mathrm{\Gamma }}_n$ for some $q\in {\mathbb{R}}^n$, where $c_k$ are real numbers not necessary to be non-negative and ${\mathrm{\Gamma }}_n$ 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–Ampère 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 $q+{\mathrm{\Gamma }}_n$ for some $q\in {\mathbb{R}}^n$, which is a way to determine the convexity of this level set.