The distance function to a finite set is a topological Morse function

TL;DR

This paper proves that the distance function to any finite set in Euclidean space is a topological Morse function, characterizes its critical points, and relates them to differential critical points, regardless of the set’s position.

Contribution

It establishes that the distance function to finite sets is topologically Morse and provides a detailed characterization of its critical points and indices.

Findings

Distance function is a topological Morse function for any finite set.

Critical points are precisely characterized and related to differential critical points.

Results hold regardless of the set’s general position.

Abstract

In this short note, we show that the distance function to any finite set $X\subset \mathbb{R}^n$ is a topological Morse function, regardless of whether $X$ is in general position. We also precisely characterize its topological critical points and their indices, and relate them to the differential critical points of the function.