Iterations of curvature images

TL;DR

This paper investigates the iterative behavior of a class of curvature image operators related to the $L_p$ Minkowski problem, establishing convergence results under specific conditions on parameters and data symmetry.

Contribution

It extends understanding of curvature image operators by proving convergence of their iterations to fixed points for certain parameter ranges and symmetry conditions.

Findings

Iterative sequences converge in Hausdorff distance to fixed points.

Convergence occurs for specific ranges of p and symmetric data.

Fixed points solve the $L_p$ Minkowski problem with prescribed data.

Abstract

We study the iterations of a class of curvature image operators $\Lambda_p^{\varphi}$ introduced by the author in (J. Funct. Anal. 271 (2016) 2133--2165). The fixed points of these operators are the solutions of the $L_p$ Minkowski problems with the positive continuous prescribed data $\varphi$. One of our results states that if $p\in (-n,1)$ and $\varphi$ is even, or if $p\in (-n,-n+1]$, then the iterations of these operators applied to suitable convex bodies sequentially converge in the Hausdorff distance to fixed points.