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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.

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  • $\begingroup$ Does your trap for a cone also trap a thin spindle? Say, two tall, base-to-base cones? $\endgroup$ Dec 23, 2020 at 15:01
  • $\begingroup$ @JosephO'Rourke: Yes. More precisely, some of the traps for the cone also trap the spindle, no matter how thin it is. $\endgroup$ Dec 24, 2020 at 4:04
  • $\begingroup$ The paper How to cage an egg by Oded Schramm (Inventiones Math. 107 (1992), 543-560) comes to mind as potentially relevant, although the question he answers is a bit different than yours. $\endgroup$
    – Dan Romik
    Dec 30, 2020 at 5:39
  • $\begingroup$ @DanRomik Romik : Yes, it is quite a bit different. $\endgroup$ Jan 12, 2021 at 0:21

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