On the convexity of general inverse σk equations

Abstract

We prove that if a level set of a degree n general inverse σk equation f(λ1,⋯,λn):=λ1⋯λn−∑k=0n−1ckσk(λ)=0 is contained in q+Γn for some q∈Rn, where ck are real numbers not necessary to be non-negative and Γn 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 σk 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 σk equation is contained in q+Γn for some q∈Rn, which is a way to determine the convexity of this level set.