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 $\mathcal{P}$.  

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>**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***.