On factorizations of smooth nonnegative matrix-values functions and on smooth functions with values in polyhedra

TL;DR

This paper explores how smooth nonnegative matrix-valued functions can be expressed as finite linear combinations with Lipschitz continuous square root coefficients, linking the problem to smooth functions with polyhedral values.

Contribution

It introduces a novel approach to representing smooth nonnegative matrix functions via fixed matrices and Lipschitz continuous coefficients, connecting to polyhedral-valued functions.

Findings

Representation of matrix functions as linear combinations with Lipschitz coefficients

Reduction of the problem to polyhedral-valued functions

Insights into smooth functions with polyhedral values

Abstract

We discuss the possibility to represent smooth nonnegative matrix-valued functions as finite linear combinations of fixed matrices with positive real-valued coefficients whose square roots are Lipschitz continuous. This issue is reduced to a similar problem for smooth functions with values in a polyhedron.