Minimal Face Numbers for Volume Rigidity

Maxwell introduced a necessary minimum number of edges in terms of the number of vertices required for a graph to yield a Euclidean rigid generic framework in ${\mathbb{R}}^3$, this count was generalised to ${\mathbb{R}}^d$, for all $d\ge 1$. In this paper, we give the analogous minimum number of $k$-simplices, for all $0\le k\le d$, required for a pure $d$-dimensional simplicial complex to yield a volume rigid generic framework in ${\mathbb{R}}^d$, for all $d\ge 1$. In order to do so, we prove some basic facts about the volume rigidity matroid and use exterior algebraic shifting, a recently added tool to the study of volume rigidity. We later prove a volume rigidity Vertex Removal Lemma and use our count to strengthen the statement.