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 $S$? 



2-neighborhood of a simplex
$\def\Vol{\mathop{\rm Vol}}$
Take $a\_1,\dots,a\_n\in S$ such that $\Vol(a\_1,\dots,a\_n)$ attains the maximum. Now, if some $b\in S$ lies outside, say, $H\_{a\_n}^+$, then $\Vol(a\_1,a\_2,\dots,a\_{n-1},b)>\Vol(a\_1,\dots,a\_n)$ which is impossible. Thus $b$ lies in the intersection of all such halfspaces.


This works for every finite set $S$.


