Stability of Order Preserving Transforms

TL;DR

This paper investigates the stability of order preserving and reversing transforms on non-negative convex functions, demonstrating that near-satisfying transforms are close to true order preserving transforms.

Contribution

It establishes stability results for order preserving/reversing transforms on convex functions, extending understanding of their structural behavior.

Findings

Transforms close to order preserving are nearly order preserving.

Stability results apply to convex functions with specific boundary conditions.

Weak conditions still imply proximity to order preserving transforms.

Abstract

The purpose of this paper is to show stability of order preserving/reversing transforms on the class of non-negative convex functions in ${\mathbb R}^n$, and its subclass, the class of non-negative convex functions attaining $0$ at the origin (these are called "geometric convex functions"). We show that transforms that satisfy conditions which are weaker than order preserving transforms, are essentially close to the order preserving transforms on the mentioned structures.