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, denote a maximal interior-disjoint set of $n$-simplices with every corner an element of $\mathcal{P}$.

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 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.