Homogeneous functions with nowhere vanishing Hessian determinant

TL;DR

This paper proves that homogeneous functions of degree between 0 and 1 with a non-zero Hessian determinant everywhere must maintain a consistent sign throughout their domain.

Contribution

It establishes a new property of such functions, showing they cannot have regions where the Hessian determinant changes sign.

Findings

Homogeneous functions of degree in (0,1) with nowhere vanishing Hessian cannot change sign.

The result constrains the possible shapes and behaviors of these functions.

It provides insight into the structure of functions with non-vanishing Hessian in geometric analysis.

Abstract

We prove that functions that are homogeneous of degree $\alpha \in (0,\,1)$ on $\mathbb{R}^n$ and have nowhere vanishing Hessian determinant cannot change sign.