The Sub-Index of Critical Points of Distance Functions

TL;DR

This paper introduces the sub-index of critical points in distance functions and explores how it influences the homotopy type of sublevel sets, providing new insights into the topology of these functions.

Contribution

It defines the sub-index of critical points and analyzes its impact on the homotopy type of sublevel sets, advancing understanding in topological data analysis.

Findings

Sub-index affects the homotopy type of sublevel sets.

Provides a new framework for analyzing critical points in distance functions.

Enhances methods for topological analysis of metric spaces.

Abstract

We define a new notion---the sub-index of a critical point of a distance function. We show how sub-index affects the homotopy type of sublevel sets of distance functions.