Convergence of an iterative scheme for the Monge–Ampère eigenvalue problem with general initial data

In this note, we revisit an iterative scheme, due to Abedin and Kitagawa (Proc. Am. Math. Soc. 148:4875–4886, 2020), to solve the Monge–Ampère eigenvalue problem on a general bounded convex domain. Using a nonlinear integration by parts, we show that the scheme converges for all convex initial data having finite and nonzero Rayleigh quotient to a nonzero Monge–Ampère eigenfunction. As an application, we obtain an energy characterization of the Monge–Ampère eigenfunctions.