On basic $r$-ball polyhedra

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 $0\le i\le d-1$, $n\ge d+1\ge 3$ 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.