On basic $r$-ball polyhedra

TL;DR

This paper introduces basic r-ball polyhedra in higher-dimensional Euclidean spaces, analyzes their face structure, and proves their global rigidity with respect to dihedral angles for dimensions greater than two.

Contribution

It defines basic r-ball polyhedra, determines maximum face counts, and proves their global rigidity in dimensions greater than two.

Findings

Maximal number of i-faces for given facets

Global rigidity with respect to dihedral angles for d>2

Face structure characterization

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.