Less Mundane Applications of the Most Mundane Functions

TL;DR

This paper explores how the simple geometric properties of linear functions can provide new insights into well-known inequalities and distance formulas, revealing their underlying structure.

Contribution

It introduces a geometric perspective on linear functions to derive and understand classical inequalities and distance measures.

Findings

Linear functions have a constant gradient field.

Geometric insights can derive classical inequalities.

Distance formulas can be understood through linear function properties.

Abstract

Linear functions are arguably the most mundane among all functions. However, the basic fact that a multi-variable linear function has a constant gradient field can provide simple geometric insights into several familiar results such as the Cauchy-Schwarz inequality, the GM-AM inequality, and some distance formulae, as we shall show.