On the Convexity of General Inverse $σ_k$ Equations

Abstract

We prove that if a level set of a degree $n$ general inverse $σ_k$ equation $f(λ_1, \cdots, λ_n) = λ_1 \cdots λ_n - \sum_{k = 0}^{n-1} c_k σ_k(λ) = 0$ is contained in $q + Γ_n$ for some $q \in \mathbb{R}^n$, where $c_k$ are real numbers not necessary to be non-negative and $Γ_n$ 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 $σ_k$ 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 $σ_k$ equation is contained in $q + Γ_n$ for some $q \in \mathbb{R}^n$, which is a way to determine the convexity of this level set.