Triangulating submanifolds: an elementary and quantified version of Whitney's method
type: Item
instance of: scholarly article
title: Triangulating submanifolds: an elementary and quantified version of Whitney's method
zbMATH Open document ID: 1468.57021
author: Jean-Daniel Boissonnat
author: Siargey Kachanovich
author: M. H. M. J. Wintraecken
published in: Discrete \& Computational Geometry
publication date: 2021-07-02T00:00:00Z
review text: \textit{S. S. Cairns} [Ann. Math. (2) 35, 579--587 (1934; Zbl 0012.03605)], \textit{L. E. J. Brouwer} [Proc. Akad. Wet. Amsterdam 42, 701--706 (1939; Zbl 0022.17201)], and \textit{H. Freudenthal} [ibid. 42, 880-901 (1939; Zbl 0022.17202)] proved that every \(C^1\) manifold admits a triangulation. Mentioning only the first paper, the authors state that ``the first proofs were complicated and not very geometric, let alone algorithmic'' (the latter being an unusual thing to say about an inutuionistic proof, Brouwer's no less). In this important paper, the authors provide a ``more quantitative/algorithmic proof'' along Whitney's ideas, for the case of \(C^2\) manifolds (with some comments on the modifications required by the \(C^{1,1}\) case) of Theorem 12A of Chapter IV in \textit{H. Whitney} [Geometric integration theory. Princeton, N. J.: Princeton University Press; London: Oxford University Press (1957; Zbl 0083.28204)], that every compact \(n\)-dimensional \(C^2\) manifold \({\mathcal M}\) embedded in \({\mathbb R}^d\) admits a triangulation. The proof uses, amongst others, the reach of \({\mathcal M}\), a perturbed version of a Coxeter triangulation of \({\mathbb R}^d\) of type \(\tilde{A}_d\), and the Delaunay protection of the Coxeter triangulation,
reviewed by: Victor V. Pambuccian
Mathematics Subject Classification ID: 57Q15
Mathematics Subject Classification ID: 57R05
zbMATH DE Number: 7367376
zbMATH Keywords: triangulations
zbMATH Keywords: manifolds
zbMATH Keywords: Coxeter triangulations
describes a project that uses: NormFet
MaRDI profile type: Publication
full work available at URL: s00454-020-00250-8
OpenAlex ID: W2964704623
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DOI: 10.1007/S00454-020-00250-8
description: scientific article; zbMATH DE number 7367376