PreprintarXiv licencearXiv
Critical Point Theory for the Distance Function on PL-Spaces
Abstract
We study the (typical) singularities of the distance function from a point in the ambient space to a polytopal complex in Euclidean space. By means of the Goresky-MacPherson splitting we generalize the existing linear theory to the quadratic distance function. Critical points occur on `selected' strata of the PL-space, where normal Morse Data determine the change of topology. On the homology level there exist indices with multiplicities, combined in a Morse vector. We focus mainly on PL-manifolds. We give bounds, depending on the dimension of the strata. In low dimensions we describe singularity types. At the end we discuss relations with Discrete Morse Theory.
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BibTeX
@misc{siersma2026critical,
title = {{Critical Point Theory for the Distance Function on PL-Spaces}},
author = {Dirk Siersma},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.13297},
eprint = {2610.13297},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.13297},
}
RIS
TY - UNPB
TI - Critical Point Theory for the Distance Function on PL-Spaces
AU - Dirk Siersma
PY - 2026
DA - 2026/10/09
PB - arXiv
DO - 10.48550/arxiv.2610.13297
UR - https://arxiv.org/abs/2610.13297
AB - We study the (typical) singularities of the distance function from a point in the ambient space to a polytopal complex in Euclidean space. By means of the Goresky-MacPherson splitting we generalize the existing linear theory to the quadratic distance function. Critical points occur on `selected' strata of the PL-space, where normal Morse Data determine the change of topology. On the homology level there exist indices with multiplicities, combined in a Morse vector. We focus mainly on PL-manifolds. We give bounds, depending on the dimension of the strata. In low dimensions we describe singularity types. At the end we discuss relations with Discrete Morse Theory.
ER -