A note on quasi-convex functions

TL;DR

This paper provides an example of smooth quasi-convex functions in three dimensions that cannot be derived from convex smooth functions via monotone smooth transformations, highlighting limitations in the relationship between these function classes.

Contribution

It introduces a specific example demonstrating that not all smooth quasi-convex functions are obtainable from convex functions through monotone smooth mappings.

Findings

Existence of smooth quasi-convex functions not representable as images of convex functions.

Illustration of limitations in transforming convex functions into quasi-convex functions.

Highlights the complexity of the relationship between convexity and quasi-convexity in smooth functions.

Abstract

We present an example of smooth quasi-convex functions in the positive octant of $\mathbb{R}^{3}$ which cannot be obtained as the images of convex smooth functions under a monotone smooth mappings of $\mathbb{R}$.