Convergence of an iterative scheme for the Monge-Amp\`ere eigenvalue problem with general initial data

In this note, we revisit an iterative scheme, due to Abedin and Kitagawa (Inverse Iteration for the Monge-Amp`ere Eigenvalue Problem, Proc. Amer. Math. Soc. 148 (2020), no. 11, 4875–4886), to solve the Monge-Amp`ere 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-Amp`ere eigenfunction. As an application, we obtain an energy characterization of the Monge–Amp`ere eigenfunctions.