In $\mathbb{R}^n$, let $K$ be a convex body and $T$ a finite set of points disjoint from the interior of $K$. Say that $T$ *traps* $K$ if there is no continuous motion of $K$ carrying $K$ arbitrarily far away from its original position during which no point of $T$ penetrates the interior of $K$. Is there a finite number $k=k(n)$ such that each $n$-dimensional convex body $K$ can be trapped by a set of at most $k$ points? If so, what is the smallest such $k$? Observe that the $3$-cube can be trapped by six points, but not by five (for the $n$-cube it's $2n$ points, but not $2n-1$). The $n$-ball can be trapped by $n+1$ points, and it seems that no $n$-dimensional convex body can be trapped by $n$ points.

The main question is: **Can every convex body in $\mathbb{R}^3$ be trapped by six points?**

More generally: **Can every convex body in $\mathbb{R}^n$ be trapped by $2n$ points?**

Footnote 1. Here is a variation of the problem, perhaps easier to handle: restrict the motions of $K$ to parallel translations.

Footnote 2. For $n=2$, a closely related problem, namely of immobilizing the body with a finite set of points, has been studied and solved: four points always suffice. Reference will be provided upon request (I will have to look it up).