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Let the simplex cover of a finite set $\mathcal{P}\subset \mathbb{E}^n,\ n\,\le\, k:=\operatorname{card}(\mathcal{P})\,\lt\infty$ of points in Euclidean $n$-space of which no $n+1$ are co-hyperplanar, denote a maximal interior-disjoint set of $n$-simplices with every corner an element of $\mathcal{P}$.


Questions:

letting $\mathrm{H}_\Sigma$ denote the set of simplex-sides, that are not shared by two simplices,

  • is $\mathrm{H}_\Sigma$ an invariant after a change of the simplex covers of a fixed pointset $\mathcal{P}$?
  • is $\mathrm{H}_\Sigma$ identical to the set of facets of the convex hull $\mathrm{CH}(\mathcal{P})$?

The reason for asking is the idea to base the definition of geometric hulls of discrete pointsets on the simplex-faces that are not shared by simplices by certain "regular" simplex-coverings of a pointset, provided the set of those undhared sides is invariant among all admissible coverings.

As there seem to be different definitions of the convex hull of a finite set of points in Euclidean spaces, in this context it shall mean what convex hull algorithms of computational geometry determine, namely the set of faces that constitute to the boundary of the intersection of all closed half-spaces in which $\mathcal{P}$ is contained.

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  • $\begingroup$ Is $H_\Sigma$ really the set of simplex facets (a set of subsets of $\Bbb E^n$), or the union of these? In the former case, I think already the cube can be decomposed in several different ways into simplices. And just a minor note: in the latter case, you probably meant to ask whether $H_\Sigma$ is the boundary of the convex hull, as the union of the simplex facets is not convex (but the convex hull is). $\endgroup$
    – M. Winter
    Feb 22, 2020 at 10:01
  • $\begingroup$ @M.Winter thanks for the feedback; I have improved my question based on it. In one point I however disagree: the convex hull itself is the set of boundary points of the set of all closed half-spaces that contain $\mathcal{P}$ and, whereas that intersection is convex, it's boundary is not. $\endgroup$ Feb 22, 2020 at 10:51
  • $\begingroup$ I see. It might be worthwhile to include your definition of convex hull into the question. I assumed the one from Wikipedia, that is, the convex hull as the smallest convex set containing these points. $\endgroup$
    – M. Winter
    Feb 22, 2020 at 10:57
  • $\begingroup$ @M.Winter thanks again for your valuable feedback! $\endgroup$ Feb 22, 2020 at 11:30

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