# Can every 3-dimensional convex body be trapped in a tetrahedral cage?

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.

• Does your trap for a cone also trap a thin spindle? Say, two tall, base-to-base cones? Dec 23, 2020 at 15:01
• @JosephO'Rourke: Yes. More precisely, some of the traps for the cone also trap the spindle, no matter how thin it is. Dec 24, 2020 at 4:04
• 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. Dec 30, 2020 at 5:39
• @DanRomik Romik : Yes, it is quite a bit different. Jan 12, 2021 at 0:21