A 4D Counter-Example Showing that DWCness Does Not Imply CWCness in n-D

From LRDE

Abstract

In this paper, we prove that the two flavours of well-composedness called Continuous Well-Composedness (shortly CWCness), stating that the boundary of the continuous analog of a discrete set is a manifold, and Digital Well-Composedness (shortly DWCness), stating that a discrete set does not contain any critical configuration, are not equivalent in dimension 4. To prove this, we exhibit the example of a configuration of 8 tesseracts (4D cubes) sharing a common corner (vertex)which is DWC but not CWC. This result is surprising since we know that CWCness and DWCness are equivalent in 2D and 3D. To reach our goal, we use local homology.

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Bibtex (lrde.bib)

@InProceedings{	  boutry.20.iwcia2,
  author	= {Nicolas Boutry and Rocio Gonzalez-Diaz and Laurent Najman
		  and Thierry G\'eraud},
  title		= {A {4D} Counter-Example Showing that {DWCness} Does Not
		  Imply {CWCness} in $n$-{D}},
  booktitle	= {Combinatorial Image Analysis: Proceedings of the 20th
		  International Workshop, IWCIA 2020, Novi Sad, Serbia, July
		  16--18, 2020},
  year		= 2020,
  editor	= {T. Lukic and R. P. Barneva and V. Brimkov and L. Comic and
		  N. Sladoje},
  volume	= {12148},
  series	= {Lecture Notes in Computer Science},
  pages		= {73--87},
  publisher	= {Springer},
  abstract	= {In this paper, we prove that the two flavours of
		  well-composedness called Continuous Well-Composedness
		  (shortly CWCness), stating that the boundary of the
		  continuous analog of a discrete set is a manifold, and
		  Digital Well-Composedness (shortly DWCness), stating that a
		  discrete set does not contain any critical configuration,
		  are not equivalent in dimension 4. To prove this, we
		  exhibit the example of a configuration of 8 tesseracts (4D
		  cubes) sharing a common corner (vertex), which is DWC but
		  not CWC. This result is surprising since we know that
		  CWCness and DWCness are equivalent in 2D and 3D. To reach
		  our goal, we use local homology.}
}