Can every 3-dimensional convex body be trapped in a tetrahedral cage?
Although the question is fairly unambiguous, I give all relevant definitions:
$\bullet$ A subset $C$ of $\mathbb{R}^n$ is an $n$-dimensional convex body if $C$ is convex, compact, and has non-empty interior.
$\bullet$ A polyhedral cage $P^{(1)}$ in $\mathbb{R}^3$ is the union of all edges (i.e., the 1-skeleton) of a convex $3$-dimensional polyhedron $P$. In particular, a tetrahedral cage is the union of the six edges of some tetrahedron.
$\bullet$ A convex $3$-dimensional body $C$ is trapped by the tetrahedral cage $T^{(1)}$, that is, by the $1$-skeleton of the tetrahedron $T$, if the cage is fixed (motionless), and if for every continuous rigid motion (rotations allowed) $f_t(C);\ 0\le t\le 1$, either $f_t(C)$ intersects $T$ for every $t\in [0,1]$ or $T^{(1)}$ contains an interior point of $f_t(C)$ for some $t_0\in [0,1]$.
In other words, $C$ cannot be moved arbitrarily far from $T$ while, during the entire motion, avoiding the cage's bars penetrating $C$'s interior.
$\bullet$ Remark. In dimension $n>3$, one can consider analogous questions, with a variety of types of a simplicial cage, by taking the $i$-skeleton of a convex $n$-dimensional simplex, with $1\le i\le n-2$.
A trivial example: A ball is trapped in the cage consisting of the $1$-skeleton of the regular tetrahedron edge-tangent to the ball. Also, obviously, every convex body can be trapped in some polyhedral cage.
For an interested reader, I suggest a few of somewhat less trivial exercises: each of the following convex bodies can be trapped in a tetrahedral cage: the cube, the circular cylinder of any (finite) height, the circular cone, the regular tetrahedron.
