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

Abstract

In this paper, we prove that the two flavors of well-composedness called Continuous Well-Composedness (shortly CWCness) and Digital Well-Composedness (shortly DWCness) are not equivalent in dimension 4 thanks to an 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 prove this new result, local (and then relative) homology are used. This paper has been submitted to IWCIA.

Continuous Well-Composedness Implies Digital Well-Composedness in n-D

“…By gathering the properties relative to well-composedness coming from [7] and from the current paper, we can see that we obtain:…”

“…DWCness and CWCness are known to be equivalent in 2D and in 3D [4,16]. As the sequel of [7] where we prove thanks to a counter-example that DWCness does not imply CWCness in 4D, we prove in this paper that CWCness implies DWCness in n-D. Some other flavors of well-composednesses exist like well-composedness in the Alexandrov sense [20,2,9,8], well-composedness on arbitrary grids [23,2], weak wellcomposedness [5], or Euler well-composedness [6], but we will not go further into details here.…”

Euler Well-Composedness

“…In 3D, only AWCness, CWCness, DWCness are equivalent. Note that no link between AGWCness and the other flavours of well-composedness were known in nD, and for n ≥ 4 the equivalences between the different flavours of well-composedness (AWCness, CWCness, DWCness and EWCness) have not been proved yet (except that AWCness implies DWCness, see [7] and that DWCness implies EWCness, see [3]) Recently, in [6], a counterexample has been given to prove that DWCness does not imply CWCness, what is an important result since it breaks with the idea that all the flavours of well-composedness are equivalent.…”