A Brunn--Minkowski inequality for the Hessian eigenvalue in convex domain

We use the deformation methods to obtain the strictly log concavity of solution of a class Hessian equation in bounded convex domain in ${\mathbb{R}}^n$, as an application we get the Brunn–Minkowski inequality for the Hessian eigenvalue and characterize the equality case in bounded strictly convex domain in ${\mathbb{R}}^n$.