On basic $r$-ball polyhedra

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.