A Geometric Lower Bound Theorem

We resolve a conjecture of Kalai relating approximation theory of convex bodies by simplicial polytopes to the face numbers and primitive Betti numbers of these polytopes and their toric varieties. The proof uses higher notions of chordality. Further, for C^2-convex bodies, asymptotically tight lower bounds on the g-numbers of the approximating polytopes are given, in terms of their Hausdorff distance from the convex body.Comment: 26 pages, 6 figures, to appear in Geometric and Functional Analysi

Rigidity and volume preserving deformation on degenerate simplices

Given a degenerate $(n+1)$-simplex in a $d$-dimensional space $M^d$ (Euclidean, spherical or hyperbolic space, and $d\geq n$), for each $k$, $1\leq k\leq n$, Radon's theorem induces a partition of the set of $k$-faces into two subsets. We prove that if the vertices of the simplex vary smoothly in $M^d$ for $d=n$, and the volumes of $k$-faces in one subset are constrained only to decrease while in the other subset only to increase, then any sufficiently small motion must preserve the volumes of all $k$-faces; and this property still holds in $M^d$ for $d\geq n+1$ if an invariant $c_{k-1}(\alpha^{k-1})$ of the degenerate simplex has the desired sign. This answers a question posed by the author, and the proof relies on an invariant $c_k(\omega)$ we discovered for any $k$-stress $\omega$ on a cell complex in $M^d$. We introduce a characteristic polynomial of the degenerate simplex by defining $f(x)=\sum_{i=0}^{n+1}(-1)^{i}c_i(\alpha^i)x^{n+1-i}$, and prove that the roots of $f(x)$ are real for the Euclidean case. Some evidence suggests the same conjecture for the hyperbolic case.Comment: 27 pages, 2 figures. To appear in Discrete & Computational Geometr

Affine stresses, inverse systems, and reconstruction problems

A conjecture of Kalai asserts that for $d\geq 4$, the affine type of a prime simplicial $d$-polytope $P$ can be reconstructed from the space of affine $2$-stresses of $P$. We prove this conjecture for all $d\geq 5$. We also prove the following generalization: for all pairs $(i,d)$ with $2\leq i\leq \lceil \frac d 2\rceil-1$, the affine type of a simplicial $d$-polytope $P$ that has no missing faces of dimension $\geq d-i+1$ can be reconstructed from the space of affine $i$-stresses of $P$. A consequence of our proofs is a strengthening of the Generalized Lower Bound Theorem: it was proved by Nagel that for any simplicial $(d-1)$-sphere $\Delta$ and $1\leq k\leq \lceil\frac{d}{2}\rceil-1$, $g_k(\Delta)$ is at least as large as the number of missing $(d-k)$-faces of $\Delta$; here we show that, for $1\leq k\leq \lfloor\frac{d}{2}\rfloor-1$, equality holds if and only if $\Delta$ is $k$-stacked. Finally, we show that for $d\geq 4$, any simplicial $d$-polytope $P$ that has no missing faces of dimension $\geq d-1$ is redundantly rigid, that is, for each edge $e$ of $P$, there exists an affine $2$-stress on $P$ with a non-zero value on $e$.Comment: 21 pages. Added a few remarks and examples (see Remark 3.6, Examples 2.5 and 3.3-3.5). To appear in IMR