Keywords: rigidity, volume, hypergraph, simplicial complex, simplicial manifold

MSC classification: 52C25 (Primary); 05E45, 57Q15 (Secondary)

1 Introduction↩︎

We will consider the general problem of deciding when a given realisation \(p\) of a hypergraph \(H=(V,E)\) in \({\mathbb{R}}^d\) is volume rigid: we consider a map \(p:V\to {\mathbb{R}}^d\) and ask if every continuous motion of the points \(p(v)\) which preserves the volume of every hyperedge of \(H\), results in a realisation which is congruent to \(p\). We will primarily be concerned with generic realisations, i.e. realisations \(p\) such that the multiset of coordinates of the points \(p(v)\), \(v\in V\), is algebraically independent over \({\mathbb{Q}}\). In this case we will see that the volume rigidity of \((H,p)\) is determined completely by the underlying hypergraph \(H\).

The special case of this problem when \(H\) is a graph is a well studied and active area of discrete geometry but the general case of hypergraphs is largely unexplored. Classical results of Cauchy [1] and Dehn [2] imply that the 1-skeleton of a convex simplicial polyhedron \(P\) is (volume) rigid in \({\mathbb{R}}^3\) i.e. every continuous motion of the vertices of \(P\) which preserves the edge lengths results in a polyhedron which is congruent to \(P\). This result was extended to 1-skeletons of convex simplicial polytopes in \({\mathbb{R}}^d\) for all \(d\geq 3\) by Whiteley [3], and to generic realisations of 1-skeletons of simplicial \((d-1)\)-manifolds in \({\mathbb{R}}^{d}\) by Kalai [4] for \(d\geq 4\) and Fogelsanger [5] for \(d\geq 3\). We will extend Kalai’s result to volume rigidity:

We will verify Theorem 1 in Section 4, using a relatively short inductive argument based on Kalai’s Theorem and Lemma 2 below (which gives the special case of Theorem 1 when \(H\) is a complete uniform hypergraph).

We conjecture that Theorem 1 remains true when \(k=d-2\).

Conjecture 1 holds when \(d=3\) by the above mentioned result of Fogelsanger. We will show that our conjecture also holds when either \(H\) is a complete \((d-1)\)-uniform hypergraph on at least \(d+1\) vertices (Theorem 2), or \(d\in \{4,5,6\}\) (Theorem 5). These results require substantial effort. In particular we will need a non-trivial extension of Whiteley’s Vertex Splitting Lemma to volume rigidity, which we prove in Section 3.2.

We also extend Whiteley’s coning lemma to the setting of volume rigidity. While this is not needed to prove our main theorems we include it since it provides a new and non-trivial extension of the classical coning theorem for graph rigidity.

In view of the above results and conjecture, it is natural to ask about the volume rigidity of generic realisations of the \(d\)-uniform hypergraph consisting of the \((d-1)\)-faces of a connected simplicial \((d-1)\)-manifold in \({\mathbb{R}}^d\). It is not difficult to see that such a hypergraph may not have enough \((d-1)\)-faces to be generically volume rigid in \({\mathbb{R}}^d\). For example, the hypergraph consisting of the 2-faces of a triangulation of the sphere is not generically volume rigid in \({\mathbb{R}}^3\). This follows since every such triangulation on \(n\) vertices has \((2n-4)\) 2-faces and hence every generic realisation in \({\mathbb{R}}^3\) will have at least \(3n-(2n-4)=n+4\) degrees of freedom. But to be rigid in \({\mathbb{R}}^3\) it would have to have precisely \(6\) degrees of freedom (given by the 6-dimensional space of translations and rotations of \({\mathbb{R}}^3\)).

We can also ask about the volume rigidity of generic realisations of the \((d+1)\)-uniform hypergraph \(H=(V,E)\) consisting of the \(d\)-faces of a connected simplicial \(d\)-manifold in \({\mathbb{R}}^{d}\). In this case Borcea and Streinu [6] observe that such a hypergraph will never be generically volume rigid in our setting since it will always have at least \(d^2+d-1\) degrees of freedom (and would have to have \({d+1\choose2}\) degrees of freedom to be generically rigid in \({\mathbb{R}}^{d}\) in our setting). On the other hand, there are several results which show that certain families of \((d+1)\)-uniform hypergraphs are ‘volume rigid’ in \({\mathbb{R}}^d\) in the sense that their generic realizations have precisely \(d^2+d-1\) degrees of freedom. Notable examples are the results of Whiteley, see [6], and Bulavka, Nevo, and Peled [7] which imply that the \(3\)-uniform hypergraph consisting of the 2-faces of a triangulated surface of low genus is ‘volume rigid’ in \({\mathbb{R}}^2\). The problem of counting the number of distinct realisations of a ‘volume rigid’ \((d+1)\)-uniform hypergraph in \({\mathbb{R}}^d\) which are equivalent to a given generic realisation is also explored in [6], [8]. In addition, Lubetzky and Peled [9] use this version of volume rigidity to analyse bootstrap percolation on random hypergraphs.

2 Volume rigidity of hypergraphs↩︎

In this section we shall give a formal definition of volume rigidity and derive some basic properties.

2.1 Volume Preserving Affine Maps↩︎

In order to extend the notion of edge-length rigidity to hyperedge-volume rigidity, we first need to understand which affine maps preserve volumes of simplices in \({\mathbb{R}}^d\). We will use the exterior algebra of \({\mathbb{R}}^d\) to accomplish this. The material in this subsection is classical and well-known, but we will provide a self-contained exposition since it is fundamental to our understanding of volume rigidity.

We use \({\mathbb{R}}^d\) to denote \(d\)-dimensional Euclidean space i.e. the \(d\)-dimensional vector space over \(\mathbb{R}\) equipped with the Euclidean inner product \(\langle \cdot, \cdot\rangle\). Let \({\boldsymbol{e}}_1,\dots, {\boldsymbol{e}}_d\) be the canonical basis of \(\mathbb{R}^d\). For an integer \(k\) with \(1\leq k\leq d\), the \(k\)-th exterior product of vectors \(x_1,\dots, x_k\in \mathbb{R}^d\) is denoted by \(x_1\wedge \dots \wedge x_k\). The \(k\)-th exterior power \(\bigwedge^k \mathbb{R}^d\) of \(\mathbb{R}^d\) is the linear span of the \(k\)-th exterior products of all possible collections of \(k\) vectors in \(\mathbb{R}^d\). An element of \(\bigwedge^k \mathbb{R}^d\) is said to be decomposable if it can be written as \(x_1\wedge \dots \wedge x_k\) for some \(x_1,\dots, x_k\in \mathbb{R}^d\).

The inner product of \(\mathbb{R}^d\) induces an inner product on \(\bigwedge^k \mathbb{R}^d\), which is given by \[\label{eq:inner95product95main} \langle x_1\wedge\dots \wedge x_k, y_1\wedge \dots \wedge y_k \rangle =\det \begin{bmatrix} \langle x_1,y_1\rangle & \langle x_2, y_1\rangle & \ldots & \langle x_k,y_1\rangle \\ \langle x_1, y_2\rangle & \langle x_2, y_2\rangle & & \vdots \\ \vdots & & \ddots & \vdots \\ \langle x_1, y_k\rangle & \dots & \dots & \langle x_k, y_k\rangle \end{bmatrix}\tag{1}\] and is then extended linearly to all elements of \(\bigwedge^k \mathbb{R}^d\).

We may use this inner product to give a concise expression for the volume of a simplex in \({\mathbb{R}}^d\). Given \(k \geq 1\) and a set of \(k+1\) points \(P=\{p_1, p_2, \dots, p_{k+1}\}\) in \(\mathbb{R}^d\) we put \[\label{eq:vol} \mathop{\mathrm{Vol}}(P)=\frac{1}{k!}\|(p_2-p_1)\wedge (p_3-p_1)\wedge \dots \wedge (p_{k+1}-p_1)\|,\tag{2}\] where \(\| \cdot \|\) denotes the norm defined by the inner product \(\langle \cdot, \cdot\rangle\) on \(\bigwedge^k {\mathbb{R}}^d\). When \(k\leq d\), \(\mathop{\mathrm{Vol}}(P)\) is the square of the volume of the simplex defined by \(P\) in any \(k\)-dimensional affine subspace of \({\mathbb{R}}^d\) which contains \(P\). Note that \(x_1\wedge\dots \wedge x_k={\boldsymbol{0}}\) whenever the multiset \(\{x_1,x_2,\ldots,x_k\}\) is linearly dependent and hence \(\mathop{\mathrm{Vol}}(P)=0\) whenever either \(k>d\), or, \(k\leq d\) and the points in \(P\) are not affinely independent.

We next turn to the problem of determining which affine maps preserve \(\mathop{\mathrm{Vol}}(P)\) for all sets \(P\) of \(k+1\) points in \({\mathbb{R}}^d\). We may assume that \(k\leq d\) since \(\mathop{\mathrm{Vol}}(P)=0\) whenever \(k>d\). In this case we may use Plücker coordinates (defined below) to express the inner product as the ordinary scalar product in a Euclidean space.

Consider the \({d\choose k}\)-dimensional Euclidean space \(\mathbb{R}^{d\choose k}\) whose coordinates are indexed by the \(k\)-subsets of \(\{1,\dots, d\}\). Let \([d]=\{1,\dots, d\}\) and denote by \({[d] \choose k}\) the set of all \(k\)-subsets of \([d]\). For clarity of the subsequent discussion, we will always assume that the coordinates of \(\mathbb{R}^{d\choose k}\) are ordered in the lexicographical ordering of the \(k\)-sets \(I\in {[d]\choose k}\). For example, if \(d=4\) and \(k=2\), then the coordinates are indexed in the order \(12, 13, 14, 23, 24, 34\).

The Plücker coordinate of an element of \(\bigwedge^k \mathbb{R}^d\) is the vector in \(\mathbb{R}^{d\choose k}\) given by taking the coordinate vector of this element with respect to the canonical basis \(\{{\boldsymbol{e}}_{i_1}\wedge \dots \wedge {\boldsymbol{e}}_{i_k}: I=\{i_1,\dots, i_k\}\in {[d]\choose k}\}\) of \(\bigwedge^k \mathbb{R}^d\). Given a decomposable element \(x_1\wedge \dots \wedge x_k\in \bigwedge^k \mathbb{R}^d\), let \(X\) be the \(d\times k\) matrix whose \(i\)-th column is \(x_i\), and let \(X_I\) be the square submatrix of \(X\) obtained by picking the rows indexed by \(i_1,\dots, i_k\) for \(I=\{i_1,\dots, i_k\}\in {[d]\choose k}\). Then, the \(I\)-th Plücker coordinate of \(x_1\wedge \dots \wedge x_k\) is \(\det X_I\). The Cauchy-Binet formula now implies that, for \(x_1\wedge \dots \wedge x_k\) and \(y_1\wedge \dots \wedge y_k\) in \(\bigwedge^k \mathbb{R}^d\), \[\begin{align} \langle x_1\wedge\dots \wedge x_k, y_1\wedge \dots \wedge y_k \rangle =\det(X^{\top} Y) =\sum_{I\in \binom{[d]}{k}} \det X_I \det Y_I, \end{align}\] and the latter term is indeed the scalar product of the Plücker coordinates of \(x_1\wedge \dots \wedge x_k\) and \(y_1\wedge \dots \wedge y_k\) in \(\mathbb{R}^{d\choose k}\). This correspondence can be extended to inner products of all pairs of elements of \(\bigwedge^k \mathbb{R}^d\) by linearity.

Given an \(m\times n\) matrix \(A\) and an integer \(k\) with \(1\leq k\leq \min\{m,n\}\), the \(k\)-th compound matrix \(A^{(k)}\) is the \({{m}\choose{k}}\times {{n}\choose{k}}\) matrix whose rows and columns are indexed by the lexicographic orderings of \({[m] \choose k}\) and \({[n]\choose k}\), respectively, and whose entries are the corresponding \(k\times k\) minors of \(A\). The compound matrix encodes the Plücker coordinates of all possible exterior products of \(k\) vectors chosen from the \(n\) column vectors of \(A\). Indeed, for \(I=\{i_1, \dots, i_k\}\) with \(i_1 < \dots < i_k\), the \(I\)-th column of \(A^{(k)}\) is \(a_{i_1}\wedge \dots \wedge a_{i_k}\), where \(a_{i_j}\) denotes the \(i_j\)-th column of \(A\). From this, one can easily check that \[Ax_1\wedge \dots \wedge Ax_k = A^{(k)} (x_1\wedge \dots \wedge x_k)\] for any \(x_1,\dots, x_k\in \mathbb{R}^n\).

We will need the following well known properties of compound matrices, see for example [10].

We can now determine which affine maps preserve \(\mathop{\mathrm{Vol}}(P)\) for all sets \(P\) of \(k\) points in \({\mathbb{R}}^d\) when \(k\leq d\). We note, for a reader who is unfamiliar with volumes of simplices, that this problem is not completely trivial since the situation for 1-simplices is different to that for \(d\)-simplices when \(d\geq 2\). Indeed, if \(P=\{p_1,\dots, p_{d+1}\}\), then \(d!{\rm Vol}(P)\) is the absolute value of the determinant of the \(d\times d\) matrix with columns \(p_2-p_1, \dots, p_{d+1}-p_1\). Hence, a map \(p\mapsto Ap+t\) for some \(A\in \mathbb{R}^{d\times d}\) and \(t\in \mathbb{R}^d\) preserves the volume of each \(d\)-simplex if and only if \(|\det A|=1\). This in particular implies that the space of such affine maps is \((d^2+d-1)\)-dimensional, see [6], [7], [9] for more details. On the other hand, it is well known that the map \(p\mapsto Ap+t\) preserves the volume of each \(1\)-simplex in \({\mathbb{R}}^d\), i.e. the length of all line segments in \({\mathbb{R}}^d\), if and only if \(A\) is orthogonal and hence the dimension of the space of all affine maps which preserve the volume of all 1-simplices in \({\mathbb{R}}^d\) is \({d+1}\choose 2\).

Combined with the expression for the volume of a set of \(k+1\) points in \({\mathbb{R}}^d\) given by (2 ), our next proposition implies that the case \(k=1\) extends to all \(1\leq k\leq d-1\), and it is the case \(k=d\) that is exceptional.

Note that the argument in the proof of Proposition 2 can be easily adapted for the case when \(k=d\). In this case, we have \(\bigwedge^d\mathbb{R}^d\simeq \mathbb{R}\), and hence (?? ) holds for \(k=d\) if and only if \(|\det A^{(d)}|=1\). This condition is equivalent to \(|\det A|=1\) by Lemma 1.

In view of applications to volume rigidity (discussed below), it would be of interest to extend Proposition 2 to the situation when ?? holds for all \(k\)-subsets of a finite set of points. To the best of our knowledge, the following problem is still open.

2.2 Volume rigidity↩︎

In view of Equation (2 ) and Proposition 2, we may define volume rigidity as follows. Given two realisations \(p, q\) of a hypergraph \(H=(V,E)\) in \(\mathbb{R}^d\), we say that \((H,p)\) is volume equivalent to \((H,q)\) if \({\rm Vol}(p(\Delta))={\rm Vol}(q(\Delta))\) holds for all \(\Delta\in E\) and that \(p\) is congruent to \(q\) if there is an orthogonal matrix \(A\) and a vector \(t\in \mathbb{R}^d\) such that \(q(v)=A p(v)+t\) for all \(v\in V\). Then \((H,p)\) is volume rigid if there is a neighborhood \(N_p\) of \(p\) in \({\mathbb{R}}^{d|V|}\) such that, for all \(q\in N_p\) such that \((H,q)\) is volume equivalent to \((H,p)\), \(q\) is congruent to \(p\). When \(H\) is a graph, this definition coincides with the standard definition of the rigidity of the bar-joint framework \((H,p)\).

A fundamental property of bar-joint rigidity is that a realization of a complete graph is rigid as long as the points of the realization are in general position. The corresponding property does not always hold for volume rigidity. For example, if \(H\) is the complete \(d\)-uniform hypergraph on \(d+1\) vertices, then the number of hyperedges of \(H\) is too small for a generic realization of \(H\) in \({\mathbb{R}}^d\) to be volume rigid when \(d\geq 3\).

Characterising which realisations of the complete \((k+1)\)-uniform⁴ hypergraph \(K_n^{k+1}\) are volume rigid appears to be a challenging open problem. Our next result comes close to solving this problem for the special case of generic realisations.

The special case of Theorem 2 when \(k\leq n-3 \leq d-2\) will be proved in the next subsection. The general case will be derived from this special case in Section 3 using a combinatorial lemma given in the same section.

Assuming that \(n\geq d+1\), so that generic realisations of \(K_n^{k+1}\) affinely span \({\mathbb{R}}^d\), and that \(k\leq d\), we are left to consider the cases when \(k\in\{d-1,d\}\). The complete hypergraph \(K_n^{d+1}\) is not generically volume rigid in \({\mathbb{R}}^d\) for all \(n\) by the above mentioned result of Borcea and Streinu [6] or Lubetzky and Peled [9]. The complete hypergraph \(K_{n}^{d}\) does not have enough hyperedges to be generically volume rigid in \({\mathbb{R}}^d\) when \(n=d+1\), but is generically volume rigid when \(n\geq d+2\) by Theorem 6 below.

2.3 Infinitesimal volume rigidity↩︎

The \(d\)-dimensional volume rigidity map of a hypergraph \(H=(V,E)\) is the function \(f:{\mathbb{R}}^{d|V|}\to {\mathbb{R}}^{|E|}\) given by \[f(p)=(f_p(\Delta_1),f_p(\Delta_2),\ldots,f_p(\Delta_m))\] where \(E=\{\Delta_1,\Delta_2,\ldots,\Delta_m\}\) and \(f_p(\Delta_h)=\mathop{\mathrm{Vol}}( p(\Delta_h))^2\) is as defined by (2 ).

We will define the volume rigidity matrix \(R(H,p)\) of a realisation \(p\) of \(H\) in \({\mathbb{R}}^d\) to be a matrix obtained from the Jacobian of \(f\) evaluated at \(p\) by suitable scalar multipications of its rows, and will associate the rows of \(R(H,p)\) with the hyperedges of \(H\) and sets of \(d\) consecutive columns with its vertices.

Let \(\Delta=\{v_0,v_1,\ldots,v_k\}\) be a hyperedge of \(H\). For each \(0\leq i\leq k\), let \(p_i = p(v_i)\) and let \(p_i^\Delta\) be the projection of \(p_i\) onto the affine subspace of \({\mathbb{R}}^d\) spanned by \(p(\Delta-v_i)\). Then (1 ) and (2 ) give \[\mathop{\mathrm{Vol}}( p(\Delta))^2=\begin{cases} \frac{1}{k^2}\|p_i-p_i^\Delta\|^2 \mathop{\mathrm{Vol}}(p(\Delta-v_i))^2 & (k\geq 2) \\ \|p_1-p_0\|^2 & (k=1) \end{cases}\] Hence, if \(k=1\), then we can take the entry in row \(\Delta\) and columns \(v_i\) of \(R(H,p)\) to be \((p_i-p_j)\) when \(\{v_i,v_j\}=\Delta\) and to be \(\boldsymbol{0}\) when \(v_i\not\in \Delta\).

It remains to consider the case when \(|\Delta|\geq 3\). Since \(p_i^\Delta=P_{\Delta-v_i}(p_i-x)+x\) for any \(x\in \mathop{\mathrm{aff}}(p(\Delta))\), where \(P_{\Delta-v_i}\) is the matrix which projects \({\mathbb{R}}^d\) onto its \((k-1)\)-dimensional linear subspace \({\rm aff}(p(\Delta-v_i))-x\), this gives \[\begin{align} k^2 \frac{\partial \mathop{\mathrm{Vol}}( p(\Delta))^2}{\partial p_i} &=2(I-P_{\Delta-v_i})(p_i-p_i^\Delta) \mathop{\mathrm{Vol}}(p(\Delta-v_i))^2\\ &=2(p_i-p_i^\Delta) \mathop{\mathrm{Vol}}( p(\Delta-v_i))^2. \end{align}\] where the second equality follows from the fact that \(p_i-p_i^\Delta\) is orthogonal to \({\rm aff}(p(\Delta-v_i))\). Hence, when \(|\Delta|\geq 3\), we can take the entry in row \(\Delta\) and columns \(v_i\) of \(R(H,p)\) to be \[\label{eq:rigmatrix} (p_i-p_i^\Delta) \mathop{\mathrm{Vol}}( p(\Delta-v_i))^2\tag{3}\] when \(v_i\in \Delta\) and to be \(\boldsymbol{0}\) when \(v_i\not\in \Delta\).

An infinitesimal motion of \((H,p)\) is a map \(\dot{p}:V\rightarrow \mathbb{R}^d\) which belongs to the right kernel of \(R(H,p)\) (when regarded as a vector in \({\mathbb{R}}^{d|V|}\)). Since every Euclidean isometry preserves the volume of each simplex in \({\mathbb{R}}^d\), the map \(v\mapsto Sp(v)+t\) for all \(v\in V\) will be an infinitesimal motion of \((H,p)\) whenever \(S\) is a skew-symmetric \(d\times d\) matrix and \(t\in \mathbb{R}^d\). We will refer to such motions as trivial infinitesimal motions of \((H,p)\).

We say that \((H,p)\) is infinitesimally volume rigid if every infinitesimal motion of \((H,p)\) is trivial. A well-known fact from rigidity theory is that the space of trivial infinitesimal motions has dimension \({d+1\choose 2}\) if \(p(V)\) affinely spans \({\mathbb{R}}^d\). More generally, the dimension of the space of trivial infinitesimal motions is \({{d+1}\choose 2}-{d-d_p\choose 2}\) when the dimension of \({\mathop{\mathrm{aff}}} (p(V))\) is \(d_p\). Hence, \((H,p)\) is infinitesimally volume rigid if and only if \(\dim \ker R(H,p)={d+1\choose 2}-{d-d_p\choose 2}\). In particular, when \(p(V)\) affinely spans \(\mathbb{R}^d\), \((H,p)\) is infinitesimally volume rigid if and only if \[\mathop{\mathrm{rank}}R(H,p)=d|V|-\binom{d+1}{2}.\]

As noted in the last subsection, the case when \(H\) is \((d+1)\)-uniform is exceptional and there exists a stronger lower bound on the dimension of the space of infinitesimal motions of \((H,p)\) in this case. This led previous authors [6], [7], [9] to use a different definition for the infinitesimal volume rigidity of realisations of \((d+1)\)-uniform hypergraphs in \({\mathbb{R}}^d\). We refer the reader to their papers for more details.

Note that, since the volume rigidity map \(f\) is a polynomial map with rational coefficients, \(f(p)\) will be a regular value of \(f\) whenever \(p\) is generic, and hence \(f^{-1}(f(p))\) will be a manifold of dimension \(d|V|-\mathop{\mathrm{rank}}R(H,p)\). Since the submanifold generated by the trivial motions of \((H,p)\) has dimension \(d|V|-\binom{d+1}{2}+\binom{d-d_p}{2}\) by [11], this gives:

Proposition 3 allows us to define a hypergraph \(H\) as being volume rigid in \({\mathbb{R}}^d\) if some, or equivalently every, generic realisation of \(H\) in \({\mathbb{R}}^d\) is infinitesimally volume rigid.

We close this section by using Proposition 3 to verify the special case of Theorem 2 when \(n \leq d+1\). Our proof also uses a result of Lovàsz [12] which determines the spectrum of the Kneser graph \(K(n,k)\). (This is the graph with vertex set \({{[n]}\choose k}\) in which two vertices \(X,Y\) are adjacent whenever \(X\cap Y=\emptyset\).)

3 Three rigidity lemmas↩︎

We will extend three fundamental lemmas for graph rigidity to hypergraph volume rigidity and apply the first lemma to complete the proof of Theorem 2. Versions of the first two lemmas (for the special case of \((d+1)\)-uniform hypergraphs in \({\mathbb{R}}^d\)) were previously given in [6], [7].

3.1 A gluing lemma and the proof of Theorem 2↩︎

The operation of gluing together two rigid graphs to construct a larger rigid graph is a standard technique in graph rigidity theory, see [13]. It is straightforward to extend this to hypergraph volume rigidity.

3.2 A vertex splitting lemma↩︎

Given a hypergraph \(H\) and two vertices \(u,v\) of \(H\), we can transform \(H\) into a new hypergraph \(H/uv\) by deleting every hyperedge which contains \(\{u,v\}\) and replacing every hyperedge \(\Delta\) which contains \(u\) and not \(v\) by \(\Delta-u+v\) (when \(\Delta-u+v\not\in {\mathcal{S}}\)). We will say that \(H/uv\) is obtained from \(H\) by contracting \(u\) onto \(v\) and that \(H\) can be obtained from \(H/uv\) by splitting \(u\) from \(v\). Note that the contraction operation is well defined but we can obtain different hypergraphs by splitting \(u\) from \(v\), depending on our choice for the sets of hyperedges incident to \(u\) and \(v\). The graphical version of these operations was first used to study the rigidity of bar-joint frameworks by Whiteley [14], who gave sufficient conditions for vertex splitting to preserve rigidity. His vertex splitting lemmas [14] and [13] have many applications in the study of bar-joint rigidity. In particular, [14] is an essential component of Fogelsanger’s proof [5] that the 1-skeleton of a connected simplicial \((d-1)\)-manifold is generically rigid in \({\mathbb{R}}^d\) when \(d \geq 3\).

Borcea and Streinu [6] and Bulavka et al [7] give a sufficient condition for vertex splitting to preserve the property that the volume rigidity matrix of a generic realisation of a \((d+1)\)-uniform hypergraph has maximum possible rank in \({\mathbb{R}}^d\). We will adapt Whiteley’s original proof technique to obtain a similar result for an arbitrary hypergraph \(H=(V,E)\) and \(u,v\in V\). It gives a common generalisation of both of Whiteley’s original vertex splitting lemmas to hypergraphs. Our sufficient condition for vertex splitting to preserve volume rigidity is in terms of the rank of a matrix \(A_{uv}(H,p,{\boldsymbol{d}})\), defined for a given realisation \(p\) of the hypergraph \(H/uv\) obtained by contracting \(u\) onto \(v\), and a given direction \({\boldsymbol{d}}\in {\mathbb{R}}^d\).

Let \(E_u\) and \(E_v\) be the sets of hyperedges of \(H\) which contain \(u\) and \(v\), respectively. Put \(E_{uv}=E_u\cap E_v\), \(E_v^u=\{\Delta\in E_v\setminus E_u:\Delta-v+u\in E_u\}\), and \(E_u^v=\{\Delta\in E_u\setminus E_v:\Delta-u+v\in E_v\}\). Recall that for each \(x\in \Delta\in E(H/uv)\), \(p(x)^\Delta\) denotes the projection of \(p(x)\) onto the affine subspace of \({\mathbb{R}}^d\) spanned by \(p(\Delta-x)\). For each \(\Delta\in E_{uv}\) let \({\boldsymbol{d}}_\Delta\) denote the projection of \({\boldsymbol{d}}\) onto the orthogonal complement of the linear subspace of \({\mathbb{R}}^d\) spanned by \(\{p(x)-p(v):x\in \Delta-u\}\) (so, in particular, \({\boldsymbol{d}}_\Delta={\boldsymbol{d}}\) when \(|\Delta|=2\)). Finally, let \(A_{uv}(H,p,{\boldsymbol{d}})\) be the matrix with rows given by the vectors in \[\{{\boldsymbol{d}}_\Delta:\Delta\in E_{uv}\}\cup \{p(v)-p(v)^\Delta:\Delta\in E_v^u\}.\] Note that the entries of \(A_{uv}(H,p,\mathbf{d)}\) are rational functions (with coefficients in \(\mathbb{Q}\)) of the coordinates of \(p\) and \(\mathbf{d}\). This follows from the following lemma.

In our particular applications in Section 4, we verify the full rank condition on \(A_{uv}(H,q,{\boldsymbol{d}})\) given in Lemma 5 by exhibiting special realisations which have full rank. It would be useful to have a simple combinatorial condition which guaranteed that \(A_{uv}(H,q,{\boldsymbol{d}})\) had full rank.

3.3 A coning lemma↩︎

We will obtain a sufficient condition for the addition of a new vertex and a set of new hyperedges to a hypergraph which is volume rigid in \({\mathbb{R}}^d\), to result in a hypergraph which is volume rigid in \({\mathbb{R}}^{d+1}\). Our result extends Whiteley’s coning lemma for bar-joint rigidity of graphs to volume rigidity of hypergraphs.

Given a graph \(G=(V,E)\) and a vertex \(v\not\in V\), the cone of \(G\) is the graph \(G*v\) obtained by adding the new vertex \(v\) and all edges from \(v\) to \(V\). A fundamental result in bar-joint rigidity due to Whiteley [15] tells us that \(G*v\) is rigid in \(\mathbb{R}^{d+1}\) if and only if \(G\) is rigid in \(\mathbb{R}^d\). Whiteley’s proof technique for sufficiency is to start with an infinitesimally rigid realization \(p\) of \(G\) in \(\mathbb{R}^{d}\) and extend this to a realisation \(p^*\) of \(G*v\) in \({\mathbb{R}}^{d+1}\) by putting the cone vertex \(v\) on the new coordinate axis. He then sends the cone vertex to infinity along this axis and shows that the rigidity matrix of \((G*v,p^*)\) converges (after an appropriate scaling and reordering of the columns so that the first \(n\) columns correspond to the \((d+1)\)-coordinates of the entries in the standard rigidity matrix of \((G*v,p^*)\)) to the block-diagonal form \[\kbordermatrix{ & v & V \\ & I_n & 0 \\ & 0 & R(G,p) },\] where \(I_n\) denotes the identity matrix of size \(|V|=n\) and \(R(G,p)\) denotes the rigidity matrix of the original framework. This block-diagonal form enables us to confirm the existence of an infinitesimally rigid realization of \(G*v\) in \({\mathbb{R}}^{d+1}\).

We shall extend Whiteley’s proof technique to volume rigidity of hypergraphs. In this case, the top-left block of the above block-diagonal form becomes more complicated so we will begin by defining it separately.

Let \(H=(V,E)\) be a hypergraph with hyperedges of size at most \(d+1\) and \(p:V\rightarrow \mathbb{R}^d\) be a point configuration such that (the multiset) \(p(\Delta)\) is affinely independent in \(\mathbb{R}^d\) for all \(\Delta\in E\). For each \(\Delta\in E\), let \(p_{\Delta}\) be the projection of the origin to the affine span of \(p(\Delta)\). Since \(p(\Delta)\) is affinely independent, \(p_{\Delta}=\sum_{x\in \Delta}\alpha_x p(x)\) for unique scalars \(\alpha_x\in {\mathbb{R}}\) with \(\sum_{x\in \Delta}\alpha_x=1\). Let \(A(H,p)\) be the \(|E|\times |V|\) matrix in which the entry in row \(\Delta\in E\) and column \(x\in V\) is \(\alpha_x\) if \(x\in \Delta\) and is zero otherwise. Let \({\cal A}_d(H)\) be the row matroid of \(A(H,p)\) for any generic \(p\). This matroid is uniquely defined since, by Lemma 4, the entries of \(A(H,p)\) are rational functions of the coordinates of \(p\).

Recall that, for \(w\in V\), \(E_w\) denotes the set of hyperedges containing \(w\). We denote the hypergraph \((V-w,\{\Delta-w: \Delta\in E_w\})\) by \(H_w\). Note that \(H_w\) has \(|V|-1\) vertices so \(\mathop{\mathrm{rank}}{\mathcal{A}}_d(H_w)\leq |V|-1\).

Proof. Let \(q\) be a generic realisation of \(H\) in \({\mathbb{R}}^d\). For \(t \in \mathbb{R}\), let \(p_t: V \to \mathbb{R}^{d+1}\) be defined by \(p_t(v) = (q(v),0)\) for \(v\in V- w\) and \(p_t(w) = (q(w),t)\). We will determine the behaviour of the entries of \(R(H,p_t)\) as \(t \to \infty\). We will assume throughout this proof that \(|\Delta|\geq 3\) for all \(\Delta\in E_w\). Hyperedges of size two can be dealt with in the same way as in the coning lemma for graphs. The argument is more complicated for larger hyperedges and we believe that mixing the two arguments will distract from the main ideas in our proof.

Choose a hyperedge \(\Delta\in E_w\). Then \(q(\Delta)\) is affinely independent and hence \(q(w)^{\Delta}=\sum_{x\in \Delta-w}\alpha_x q(x)\) for unique scalars \(\alpha_x\) with \(\sum_{x\in \Delta-w}\alpha_x=1\).

Claim 4. For each \(u\in \Delta-w\), \(q(w)^{\Delta}-q(w)^{\Delta-u}=\alpha_u\left( q(u)-q(u)^{\Delta-w}\right)\).

Proof. Let \(\pi\) be the orthogonal projection of \(\mathbb{R}^d\) onto the affine span of \(q(\Delta-u-w)\). Then \(q(w)^{\Delta}-\pi(q(w)^{\Delta})=\alpha_u\left( q(u)-\pi(q(u))\right)\). The claim follows by observing that \(\pi(q(w)^{\Delta})=q(w)^{\Delta-u}\) and \(\pi(q(u))=q(u)^{\Delta-w}\). ◻

Let \(I_d\) denote the \(d \times d\) identity matrix.

Claim 5. For each \(u \in \Delta- w\), \[\lim_{t \to \infty} \left(R(H,p_t)_{\Delta,u}\begin{pmatrix} \frac{1}{t^2}I_{d} & 0 \\ 0 & -\frac{1}{t} \end{pmatrix}\right) = \left( \frac{1}{(|\Delta|-2)^2} R(H_w,q|_{V-w})_{\Delta-w,u}, \mathop{\mathrm{Vol}}(q(\Delta-w))^2 \alpha_u \right).\]

Proof. We have \(q(\triangle) \subset {\mathbb{R}}^d \subset {\mathbb{R}}^{d+1}\) where the last inclusion is induced by \(x \in {\mathbb{R}}^d \mapsto (x,0) \in {\mathbb{R}}^{d+1}\). Let \(\pi:\mathbb{R}^{d+1} \to \mathop{\mathrm{aff}}(q(\Delta-w))\) be the orthogonal projection of \(\mathbb{R}^{d+1}\) onto the affine subspace spanned by \(q(\Delta-w)\). Note that for all \(v \in \Delta-w\) and all \(t \in {\mathbb{R}}\), \(p_t(v) = (q(v),0)\). An elementary argument using similarity of triangles (see Figure 1) and the fact that \(p_t(u)_{d+1}=0\) gives \[\label{eqn:sim} {\rm sign}(\alpha_u)\frac{\| p_t(u) - \pi(p_t(u)^\Delta)\|}{ (p_t(u)^\Delta)_{d+1}} =\frac{t}{\| p_t(w)^\Delta - p_t(w)^{\Delta-u}\|},\tag{6}\] where \({\rm sign}(\alpha_u)\) denotes the sign of \(\alpha_u\), which, for large \(t\), is determined by which side of \({\rm aff}(p_t(\Delta-u))\) the point \(p_t(u)\) belongs to in \({\rm aff}(p_t(\Delta))\). Both sides of 6 are equal to the tangent of the angle between the hyperplanes \({\rm aff}(p_t(\Delta-u))\) and \({\rm aff}(p_t(\Delta-w))\) in \({\rm aff}(p_t(\Delta))\).

We also have \[\label{eqn:v}\mathop{\mathrm{Vol}}(p_t(\Delta-u))^2 = \frac{1}{(|\Delta|-2)^2}\|p_t(w) - p_t(w)^{\Delta-u}\|^2\mathop{\mathrm{Vol}}(p_t(\Delta-w-u))^2.\tag{7}\] Hence, \[\begin{align} &\frac{(R(H,p_t)_{\Delta,u})_{d+1}}{t} =\frac{\mathop{\mathrm{Vol}}(p_t(\Delta-u))^2}{t} (p_t(u)-p_t(u)^{\Delta})_{d+1} \nonumber\\ &=-{\rm sign}(\alpha_u) \frac{\mathop{\mathrm{Vol}}(p_t(\Delta-u))^2}{t^2} \|p_t(u) -\pi(p_t(u)^\Delta)\| \| p_t(w)^\Delta - p_t(w)^{\Delta-u}\| \nonumber\\ &= -{\rm sign}(\alpha_u) \frac{\mathop{\mathrm{Vol}}(q(\Delta-u-w))^2}{ t^2 (|\Delta|-2)^2} \|p_t(w) - p_t(w)^{\Delta-u}\|^2 \|p_t(u) -\pi(p_t(u)^\Delta)\| \| p_t(w)^\Delta - p_t(w)^{\Delta-u}\| \label{eqn:vol}, \end{align}\tag{8}\] where the first equation is by definition, the second equation follows from \((p_t(u))_{d+1}=0\) and (6 ), and the third equation follows from (7 ).

We now send \(t\) to \(\infty\). The fact that \(p_t(w) - (q(w),0)\) is orthogonal to \(\mathop{\mathrm{aff}}(p_t(\Delta-w))\) gives \[\label{eqn:lim} \lim_{t \to \infty} \begin{pmatrix} p_t(w)^{\Delta} \\ p_t(w)^{\Delta-u} \\ \pi(p_t(u)^{\Delta}) \end{pmatrix} \rightarrow \begin{pmatrix} (q(w)^{\Delta},0)\\ (q(w)^{\Delta-u},0)\\ (q(u)^{\Delta-u},0) \end{pmatrix}\tag{9}\] and hence \[\label{eqn:r}\lim_{t \to \infty} \frac{\|p_t(w) - p_t(w)^{\Delta-u}\|}{t} = 1.\tag{10}\] We can combine 8 , 9 and 10 to obtain \[\begin{align} &\lim_{t \to \infty} \frac{(R(H,p_t)_{\Delta,u})_{d+1}}{t} \\ &=-{\rm sign}(\alpha_u) \frac{\mathop{\mathrm{Vol}}(q(\Delta-u-w))^2}{ (|\Delta|-2)^2} \|q(u) -q(u)^{\Delta-w}\| \| q(w)^\Delta - q(w)^{\Delta-u}\| \notag\\ &= -\frac{\mathop{\mathrm{Vol}}(q(\Delta-u-w))^2}{ (|\Delta|-2)^2} \|q(u) -q(u)^{\Delta-w}\|^2 \alpha_u \qquad (\text{by Claim~\ref{clm:alpha}}) \\ &= - \mathop{\mathrm{Vol}}(q(\Delta-w))^2 \alpha_u \end{align}\] which verifies that equality holds for the \((d+1)\)-th component in the statement of the claim.

In addition we deduce that \[\begin{align} \label{eqn:o} \lim_{t \to \infty} \frac{R(H,p_t)_{\Delta,u}}{t^2} &= \lim_{t \to \infty } \frac{\mathop{\mathrm{Vol}}(p_t(\Delta-u))^2}{t^2} (p_t(u)-p_t(u)^\Delta) & \\ &= \lim_{t \to \infty } \frac{\mathop{\mathrm{Vol}}(p_t(\Delta-w-u))^2\|p_t(w) - p_t(w)^{\Delta-u}\|^2}{(|\Delta |-2)^2t^2} (p_t(u)-p_t(u)^\Delta) & \text{using (\ref{eqn:v})}\\ &= \frac{\mathop{\mathrm{Vol}}(q(\Delta-w-u))^2}{(|\Delta|-2)^2} (q(u)-q(u)^{\Delta-w},0) & \text{using (\ref{eqn:lim}) and (\ref{eqn:r})} \\ &= \left(\frac{1}{(|\Delta|-2)^2} R(H_w,q|_{V-w})_{\Delta-w,u},0\right) & \end{align}\tag{11}\] This completes the proof of the claim. ◻

Let \(R_t\) be the matrix obtained from \(R(H,p_t)\) by deleting the \((d+1)\) columns associated with \(w\) and by permuting columns so that the last \(|V|-1\) columns consist of the columns indexed by the \((d+1)\)-th coordinates of the vertices in \(V-w\). Since \(p_t(v)=(q(v),0)\) for \(v\in V-w\), \(R_t\) can be written as \[R_t=\kbordermatrix{ &\text{first d coordinates of V-w} & \text{(d+1)-th coordinates of V-w}\\ E_w & B_t & A_t \\ E\setminus E_w & R(H-w,q|_{V-w}) & 0 }\] for some \(A_t\) and \(B_t\). Since \(\mathop{\mathrm{rank}}R(H-w, q|_{V-w})=d(|V|-1)-{d+1\choose 2}\) by the hypothesis that \(H-w\) is volume rigid in \({\mathbb{R}}^d\), it suffices to show that \(\mathop{\mathrm{rank}}A_t=|V|-1\) for some \(t\).

By Claim 5, \(\lim_{t\rightarrow \infty} -\frac{1}{t} A_t =D A(H_w,q|_{V-w})\), where \(D\) denotes the \(|E_w| \times |E_w|\) diagonal matrix with diagonal entries \(\mathop{\mathrm{Vol}}(q(\Delta-w))^2\) for \(\Delta \in E_w\). Since \(q\) is in general position, \(D\) is non-singular. The hypothesis that \(\mathop{\mathrm{rank}}{\mathcal{A}}_d(H_w)=|V|-1\) now gives \(\mathop{\mathrm{rank}}D A(H_w,q|_{V-w}) =|V|-1\). Thus, for all but finitely many values of \(t\), \(\mathop{\mathrm{rank}}A_t=|V|-1\). This completes the proof. ◻

Remarks. A more careful analysis in the proof of Lemma 6 tells us that:
(a) when \(|\Delta|\geq 3\) for each \(\Delta\in E_w\), the matrix \(R_t\) satisfies \[\label{eqn:limit} \lim_{t\to \infty} \begin{pmatrix} I_{|E_w|} & 0 \\ 0 & t^2 I_{|E\setminus E_w|} \end{pmatrix}R_t \begin{pmatrix} \frac{1}{t^2}I_{d|V|} & 0 \\ 0 & \frac{1}{t} I_{|V|-1} \end{pmatrix} = \begin{pmatrix} D'R(H_w,q|_{V-w}) & DA(H_w,q|_{V-w}) \\ R(H-w,q|_{V-w}) & 0 \end{pmatrix}\tag{12}\]

where \(D,D'\) denote the \(|E_w| \times |E_w|\) diagonal matrices with diagonal entries \(\mathop{\mathrm{Vol}}(q(\Delta-w))^2\) and \((|\Delta|-2)^2\) for \(\Delta \in E_w\), respectively. Hence, even if \(H-w\) is not volume rigid in \(\mathbb{R}^d\), we may still be able to show that \(H\) is volume rigid in \(\mathbb{R}^{d+1}\) by analysing the block matrix on the right hand side of (12 ).
(b) any infinitesimally volume rigid realisation of \(H-w\) in \({\mathbb{R}}^d\) can be extended to an infinitesimally rigid realisation of \(H\) in \({\mathbb{R}}^{d+1}\) by placing \(w\) at all but finally many points on the \((d+1)\)’th coordinate axis.

We next obtain a sufficient condition for the rank of the matroid \({\mathcal{A}}_d(H)\) of a \(k\)-uniform hypergraph \(H=(V,E)\) to be equal to \(| V|\). A \(k\)-uniform hypergraph \(H\) is strongly connected if, for any two distinct \(\Delta,\Delta'\in E\), there exists a sequence of hyperedges \(\Delta_1,\Delta_2,\ldots,\Delta_m\) such that \(\Delta_1=\Delta\), \(\Delta_m=\Delta'\) and \(|\Delta_i\cap \Delta_{i+1}|=k-1\) for all \(1\leq i\leq m-1\). We will refer to the maximal strongly connected sub-hypergraphs of \(H\) as the strongly connected components of \(H\). It is easy to see that the property of belonging to the same strongly connected component is an equivalence relation on \(E\) and hence the edge sets of the strongly connected components of \(H\) partition \(E\).

4 Volume rigidity of simplicial complexes↩︎

We will prove Theorem 1 and show that Conjecture 1 holds when \(d=4,5,6\).

We first need to introduce some terminology. Recall that an (abstract) simplicial complex \({\mathcal{S}}\) is a family of sets which is closed under inclusion. We refer to the elements of \({\mathcal{S}}\) of cardinality \(k+1\) as \(k\)-faces and the maximal elements of \({\mathcal{S}}\) as facets. We use \(V({\mathcal{S}})\) and \(E({\mathcal{S}})\) to denote the set of vertices, i.e. \(0\)-faces, and edges, i.e. 1-faces, of \({\mathcal{S}}\), respectively. Given \(X\in {\mathcal{S}}\), the link of \(X\) in \({\mathcal{S}}\) is the complex \[\mathop{\mathrm{lk}}(X)= \mathop{\mathrm{lk}}_{\mathcal{S}}(X) = \{Y:X\cap Y=\emptysetandX\cup Y\in {\mathcal{S}}\}\] and the star of \(X\) is the complex \[\mathop{\mathrm{st}}(X)= \mathop{\mathrm{st}}_{\mathcal{S}}(X) = \{X'\cup Y:X'\subseteq XandY\in \mathop{\mathrm{lk}}(X)\}.\] We say \({\mathcal{S}}\) is a simplicial \(d\)-complex if each of its facets has cardinality \(d+1\).

Recall that the topological realisation of a simplicial complex \({\mathcal{S}}\) is the subset of \({\mathbb{R}}^{V({\mathcal{S}})}\) defined by \[|{\mathcal{S}}| = \bigcup_{X \in {\mathcal{S}}} \mathop{\mathrm{conv}}\{{\boldsymbol{e}}_v: v \in X\}\] where, for a vertex \(v \in V({\mathcal{S}})\), \({\boldsymbol{e}}_v\) is the corresponding standard basis vector of \({\mathbb{R}}^{V({\mathcal{S}})}\). We say that \({\mathcal{S}}\) is a (connected) simplicial \(d\)-manifold if \(|{\mathcal{S}}|\) is a connected \(d\)-manifold.

We will refer to the \((k+1)\)-uniform hypergraph whose hyperedges are the \(k\)-faces of a simplicial complex \({\mathcal{S}}\) as the \(k\)-skeleton hypergraph of \({\mathcal{S}}\) and denote it by \(H_k({\mathcal{S}})\). A simplicial \(d\)-complex is said to be strongly connected if its \(d\)-skeleton hypergraph is strongly connected.

Theorem 1 follows immediately from our next result and the above mentioned result of Kalai [4] (which tells us that the 1-skeleton of a simplicial \((d-1)\)-manifold is rigid in \({\mathbb{R}}^d\)).

Combined with Theorem 1, Conjecture 1 would imply that the \(k\)-skeleton hypergraph of a connected simplicial \((d-1)\)-manifold is volume rigid in \({\mathbb{R}}^d\) for all \(1\leq k\leq d-2\). We first point out that the same conclusion holds for strongly connected simplicial \(d\)-complexes.

We next verify Conjecture 1 when \(d=4,5,6\). Our inductive proof naturally leads us to consider a larger family of simplicial complexes than simplicial manifolds. Following Kalai [4], we define a simplicial homology \(k\)-manifold to be a simplicial \(k\)-complex \({\mathcal{S}}\) with the property that, for all \(0\leq j\leq k-1\) and all \(j\)-faces \(X\) of \({\mathcal{S}}\), the simplicial homology (with coefficients in \(\mathbb{Z}\)) of \(\mathop{\mathrm{lk}}_{\mathcal{S}}(X)\) is the same as that of a simplicial \((k-j-1)\)-sphere. We say that \({\mathcal{S}}\) is a simplicial homology \(k\)-sphere if, in addition, \({\mathcal{S}}\) has the same simplicial homology as a simplicial \(k\)-sphere. We will use the fact that every simplicial homology \(2\)-sphere is a simplicial 2-sphere.

We also need the following construction. Given a simplicial complex \({\mathcal{S}}\) and a finite set of vertices \(Z\) disjoint from \(V({\mathcal{S}})\), the simplicial cone of \({\mathcal{S}}\) with \(Z\) is the complex \[{\mathcal{S}}*Z=\{X\cup Z':X\in {\mathcal{S}} andZ'\subseteq Z\}.\]

In order to extend the above proof technique to verify Conjecture 1 for all \(d\geq 3\), we will need an analogue of Lemma 10 for all \(d\geq 4\). The inductive proof of this lemma would extend to all \(d\geq 4\) if we could verify the following conjecture as the base case of our induction.

The hypothesis that \(|V({\mathcal{S}})|\geq \lceil d/2\rceil+2\) is needed in Conjecture 7 since, if \(|V({\mathcal{S}})|\leq\lceil d/2\rceil+1\), then a straightforward calculation shows that \(|E(H_{d-2}(S*Z))| < d|V(S*Z)|-\binom{d+1}2\), so \(S*Z\) cannot be volume rigid in \(\mathbb{R}^d\). Note, however, that even if we could verify Conjecture 7, we would still need to extend the argument in Cases 1 and 2 of the proof of Theorem 5 to cover the situation when \(5\leq |V(\mathop{\mathrm{lk}}(Z))|\leq\lceil d/2\rceil+1\) in order to verify Conjecture 1.

5 Closing remarks and open problems↩︎

5.1 Volume rigidity of \(d\)-uniform hypergraphs in \({\mathbb{R}}^d\)↩︎

It seems to be difficult to find families of \(d\)-uniform hypergraphs which are volume rigid in \({\mathbb{R}}^d\). One notable exception is the following recent strengthening of Theorem 2 by Lew, Nevo, Peled, and Raz [17] to include complete \(d\)-uniform hypergraphs with at least \(d+2\) vertices.

Note that the \(k=d\) case of Theorem 6 does not extend to \((d-1)\)-skeleton hypergraphs of simplicial \((d-1)\)-manifolds since these hypergraphs may not have enough hyperedges to be volume rigid in \({\mathbb{R}}^d\). Consider, for example, the \(2\)-skeleton hypergraph of a simplicial 2-sphere in \({\mathbb{R}}^3\).

5.2 Infinitesimal volume rigidity of convex simplicial polytopes↩︎

Whiteley [3] showed that, when \(d\geq 3\), the 1-skeleton of every convex simplicial \(d\)-polytope is infinitesimally rigid in \({\mathbb{R}}^d\) (without any assumption of genericity). Could this result be extended to show that the \(k\)-skeleton of every convex simplicial \(d\)-polytope is infinitesimally volume rigid in \({\mathbb{R}}^d\) for all \(1\leq k\leq d-2\)?

Acknowledgements↩︎

We would like to thank David Ellis for his suggestion that we could use the result from [12] to simplify our proof of Lemma 2, and Alan Lew, Yuval Peled, Eran Nevo and Orit Raz for sharing a draft of their preprint [17] just before both articles were submitted to arXiv. We also thank the anonymous referees for many helpful suggestions that improved the paper, including the statement of Theorem 3. This work was supported by JST PRESTO Grant Number JPMJPR2126, JSPS KAKENHI Grant Number 20H05961, and EPSRC overseas travel grant EP/T030461/1.

References↩︎