a cross polytope

Question

Consider $2n$ positive numbers $a_{i}$ and define the set $S=\prod_{i=1}^{n}[-a_i,a_{i+n}]\subset \mathbb{R}^n.$ We pick one point $x_i$ from the interior of each facet ($n-1$-dimentional face) of S. Does the convex hull of the set $\{x_i\}_i$ has non-empty interior as subset of $\mathbb{R}^n$? (an $n$-dimensional convex set).

Answer 1

In short: no.

Imagine our old 3D cube and cut it with a plane passing through the center and perpendicular to a great diagonal. It will pass through all six faces. Pick your points somewhere within this section, and their convex hull will obviously be planar as well.

I guess this generalizes naturally to higher dimensions.