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

TL;DR

The paper proves that the distance function to any finite set $X\subset \mathbb{R}^n$, denoted $d_X$, is a topological Morse function without requiring general position. It precisely characterizes topological critical points: points in $X$ are critical of index $0$, while non-$X$ points with $z\in \mathrm{Conv}(\Pi_X(z))$ are regular if a separating vector exists in $\mathrm{Span}(\Pi_X(z)-z)$, and are critical of index $\dim(\mathrm{Span}(\Pi_X(z)-z))$ otherwise; all other points are regular. The results reveal that topological critical points form a subset of the differential critical points, and the authors provide constructive local normal forms via homeomorphisms and Morse-type reasoning to establish non-degeneracy. This advances topological data analysis by giving robust, position-agnostic Morse structure to $d_X$ for finite point clouds, with potential extensions to Morse-type function frameworks.

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.

Figures (2)

• Figure 1: The level sets of the distance function to three sets of points are represented. From left to right, the central point is a non-degenerate topological critical point of index $1$, a topological regular point and a non-degenerate topological critical point of index $2$ respectively.
• Figure 2: Three level sets of the distance function to the set $X=\{x_1,\ldots,x_4\}$ and the points $P$, $p$ and $\pi(p)$ as defined in the proof of \ref{['prop:regular_points']}.

Theorems & Definitions (11)

• Definition 1.1: Topological Morse functions morse_topologically_1959
• Theorem 1.2
• Proposition 2.1
• proof
• Proposition 2.2
• proof
• Lemma 2.3
• proof
• Lemma 2.4
• proof