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.
