The distance function to a finite set is a topological Morse function
<|reference_start|>The distance function to a finite set is a topological Morse function: 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.<|reference_end|>
@article{arnal2024the,
    title={The distance function to a finite set is a topological Morse function},
    author={Charles Arnal},
    journal={arXiv preprint arXiv:2407.15578},
    year={2024},
    archivePrefix={arXiv},
    eprint={2407.15578},
    primaryClass={math.DG cs.CG}
}