Inscribing a regular octahedron into polytopes

Abstract

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

Hom-polytopes

“…By an easy refinement of Corollary 4.3(4), we see that there are 2mn vertex maps of type (2), and mn(n − 2) of type (3). There are also n of type (1).…”

“…In the context of extremal maps from regular polytopes, it is interesting to remark that the octahedron ♦ 3 admits an affine embedding into any simple 3-dimensional polytope P so that the vertices of ♦ 3 map to the boundary ∂P [2].…”

“…Consequently, (f t ) (−1,1) must be constant on P -a contradiction. This simple observation, together with Theorem 4.1 (2), suggests that to look for non-vertex maps f : P → Q with f surj and f inj both vertices, we should consider maps that keep the vertices of Im f away from vert(Q). We first give a general construction and then construct examples of vertex factorizations of non-vertex maps.…”

“…But dim Im σ = 2 as well because no affine line can intersect all segments [w i , w i+1 ] simultaneously -this is where we use the inequality n ≥ 6. 2 In particular, Im(ι t ) ∼ = P for all t ∈ [0, 1]. Since none of the mentioned incidences is lost for t = 1 and σ is a vertex, σ belongs to more facets of Hom(P, Q) than ι.…”

A Survey on the Square Peg Problem

“…Karasev [19] generalized the proof to arbitrary odd prime powers. Akopyan and Karasev [1] proved the same theorem for n = 3 in case Γ is the boundary of a simple polytope by a careful and nontrivial limit argument from the smooth case.…”