Inscribing a regular octahedron into polytopes

TL;DR

This paper proves that every simple polytope in three-dimensional space can have a regular octahedron inscribed within it, expanding understanding of polytope inscribability.

Contribution

It establishes that all simple polytopes in three dimensions can contain an inscribed regular octahedron, including some non-simple cases, which was previously unknown.

Findings

All simple 3D polytopes admit inscribed regular octahedra.

Some non-simple polytopes also admit such inscribed octahedra.

The result broadens the class of polytopes known to contain inscribed regular octahedra.

Abstract

We prove that any simple polytope (and some non-simple polytopes) in $\mathbb R^3$ admits an inscribed regular octahedron.