# Differential Inequalities for Distance Comparison

## TL;DR

This paper generalizes a differential inequality used in Alexandrov geometry to compare 1-dimensional distance functions, enabling curvature bounds to be checked directly on smooth functions for various curvature conditions.

## Contribution

It extends the existing differential inequality for zero curvature to arbitrary curvature bounds, broadening the tools for curvature comparison in geometric analysis.

## Key findings

- Generalized differential inequality for arbitrary curvature bounds
- Enhanced methods for curvature comparison in Alexandrov geometry
- Applicable to smooth 1-dimensional distance functions

## Abstract

Comparison of $1$-dimensional distance functions is a basic tool in Alexandrov geometry and it is used to characterize spaces with curvature bounded above or below. For the zero curvature bound there is a differential inequality which enables one to check this comparison directly on a given smooth $1$-dimensional distance function. In this note we give a generalization of this property to arbitrary curvature bounds.