On basic $r$-ball polyhedra

TL;DR

This work introduces basic $r$-ball polyhedra in $\mathbb{E}^d$ ($d>1$) and develops their face-structure via farthest-point Voronoi diagrams, establishing a constructive link between the polyhedron $\mathbf{P}_{C,r}$ and its center-polytope $\mathbf{C}={\rm conv}(C)$. It proves a McMullen-style upper bound for the number of simple faces, $f_{k-1}(\mathbf{P}_{C,r}) \le c_{d-k}(n,d)$, by translating face counts to the spherical complex $\partial D_f(C)$ and invoking Stanley's Upper Bound Theorem. The paper also proves a global rigidity result for $d\ge 3$, showing that basic $r$-ball polyhedra are determined up to congruence by their face lattice and inner dihedral angles, leveraging Alexandrov’s rigidity framework via the boundary Delaunay complexes. Together, these results extend combinatorial and rigidity properties of 3D normal ball polyhedra to higher dimensions and connect geometric realizations to inscribed polyhedral complexes. The conclusions point to further exploration of standard $r$-ball polyhedra and local rigidity phenomena in higher dimensions.

Abstract

This note introduces the class of basic $r$-ball polyhedra in the $d$-dimensional Euclidean space $\mathbb{E}^{d}$ for $d>1$ and $r>0$. We investigate their face structure and, for given integers $0\leq i\leq d-1$, $n\geq d+1\geq 3$ determine the maximal number of $i$-dimensional faces among all basic $r$-ball polyhedra in $\mathbb{E}^{d}$ with $n$ facets. In addition, we establish that for $d>2$, every basic $r$-ball polyhedron is globally rigid with respect to its inner dihedral angles.

Theorems & Definitions (18)

• Definition 1.1
• Remark 1.2
• Theorem 1.3
• Remark 1.4
• Definition 1.5
• Theorem 1.6
• Theorem 2.1
• proof : Proof of Theorem \ref{['basic properties-1']}:
• Lemma 2.2
• proof : Proof of Lemma \ref{['rec1']}: