Hunting for Reduced Polytopes

Abstract

We show that there exist reduced polytopes in three-dimensional Euclidean space. This partially answers the question posed by Lassak [10] on the existence of reduced polytopes in d-dimensional Euclidean space for d ≥ 3. Moreover, we prove a novel necessary condition on reduced polytopes in threedimensional Euclidean space.

Vertex-Facet Assignments For Polytopes

“…Proof. We show, that a face σ ∈ F(P ) satisfying (i) or (ii), also satisfies (3). By Theorem 5.3 we then obtain the existence of a vertex-facet assignment.…”

“…The content of the present paper can be motivated by a metrical notion called reducedness introduced for general convex sets in [5]. While the existence of reduced polytopes is clear in the Euclidean plane, the only known examples of reduced polytopes in higher-dimensional Euclidean spaces were obtained in [3]. At the core of this existence question lies a combinatorial problem: The authors of [1,Theorem 4] show, that in a reduced polytope, each vertex is (in some sense) opposite to a non-incident facet, and this assignment is injective.…”

Vertex-facet assignments for polytopes

“…While the existence of reduced polytopes is clear in the Euclidean plane (e.g. a regular triangle), the only known examples in higher dimensions were obtained in González Merino et al (2018)-a single family of 3-dimensional polyhedra (see Fig. 1).…”