The question of whether every 3-dimensional convex body can be trapped in a tetrahedral cage involves analyzing the geometric constraints imposed by the cage (the 1-skeleton of a tetrahedron) on rigid motions of the convex body. A convex body ( C ) is trapped by a tetrahedral cage ( T^{(1)} ) if any continuous rigid motion of ( C ) either keeps it intersecting the solid tetrahedron ( T ) t...

Short answer: This appears to be open in general. No proof is known that every 3‑dimensional convex body can be trapped by the 1‑skeleton of some tetrahedron, and no counterexample is known either.

Remarks and partial results:

• Many familiar convex bodies can indeed be trapped by a tetrahedral cage, as you noted (sphere, cube, finite circular cylinder, circular cone, regular tetrahedron). These ...

Short answer
The problem is still open. No proof is known that six edges of some tetrahedron always suffice, and no explicit convex body is known that can escape from every tetrahedral 1–skeleton. All partial results obtained so far go only in one of the following two directions.

1. Positive results. One can trap many special classes of convex bodies – for instance • every ball,
   ...

Short answer
At the present time nobody knows.
No proof is available that a single tetrahedral “wire-frame’’ can always confine an arbitrary 3–dimensional convex body, and no example of a convex body that provably escapes from every tetrahedral cage is known either. The problem is open.

What is known

1. Positive examples ...

First, let me understand the definitions. A convex body in 3D is a convex, compact set with non-empty interior. A tetrahedral cage is the 1-skeleton of a tetrahedron, meaning just the six edges. Trapping means t...

This is an excellent and classic question in combinatorial and convex geometry. The answer is no.

Summary of the Answer

The answer to the question is no. There exists a 3-dimensional convex body that cannot be trapped by any tetrahedral cage. Such a body is typically long and thin, like a "fat needle" or a "bicone". The proof relies on a special geometric property of the tetrahedron's 1-...

Here is a summary of the state of the art on the MathOverflow question “Can every 3‐dimensional convex body be trapped in a tetrahedral cage?”:

1. Definitions.
   – A 3–dimensional convex body (C\subset\Bbb R^3) is a compact convex set with nonempty interior.
   – A tetrahedral cage is the 1–skeleton (T^{(1)}) of some tetrahedron (T): six straight “bars” meeting in four vertices...

Can Every 3-Dimensional Convex Body Be Trapped in a Tetrahedral Cage?

This is an interesting geometric problem about containment and motion. Let me analyze it carefully.

Understanding the Problem

A 3D convex body is "trapped" by a tetrahedral cage (the 1-skeleton of a tetrahedron, consisting of 6 edges) if the body cannot be moved arbitrarily far away without either:

1. Intersecting the ca...

This is a fascinating question about whether every 3-dimensional convex body can be trapped in a tetrahedral cage (the edge skeleton of some tetrahedron).

Let me analyze this problem:

Key Observations:

1. A tetrahedral cage consists of only 6 edges (line segments) in 3D space
2. For a convex body to be trapped, it must be impossible to move it arbitrarily far away without either:
   • Conti...