Let $\Delta$ be an $n$-simplex in ${\mathbb R}^{n-1}$. For each vertex $v$ of $\Delta$ let $H_v$ be the hyperplane through $v$ and parallel to the opposite facet. By 2-neighborhood of a simplex I mean a simplex which is the intersection of the half-spaces $H_v^+$ containing $\Delta$. My question is:

Let $S$ be a collection of $2n - 4$ distinct points on ${\mathbb R}^{n-1}$ spanning ${\mathbb R}^{n-1}$. Does there exist $n$ points in $S$ so that the $2$-neighborhood of the simplex formed by these $n$ points containing $S$?

Let $\Delta$ be an $n-1$-simplex in ${\mathbb R}^{n-1}$. For each vertex $v$ of $\Delta$ let $H_v$ be the hyperplane through $v$ and parallel to the opposite facet. By 2-neighborhood of a simplex I mean a simplex which is the intersection of the half-spaces $H_v^+$ containing $\Delta$. My question is:

Let $S$ be a collection of $2n - 4$ distinct points on ${\mathbb R}^{n-1}$ spanning ${\mathbb R}^{n-1}$. Does there exist $n$ points in $S$ so that the $2$-neighborhood of the simplex formed by these $n$ points containing $S$?