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 $e$.