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Abstract 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 $$e$$.
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On the Hypergraph Connectivity of Skeleta of Polytopes
We show that for every d-dimensional polytope, the hypergraph whose nodes are kfaces and whose hyperedges are (k +1)-faces of the polytope is strongly (d −k)-vertex connected, for each 0 ≤ k ≤ d − 1.
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- Award ID(s):
- 1855726
- PAR ID:
- 10337854
- Date Published:
- Journal Name:
- Discrete & Computational Geometry
- ISSN:
- 0179-5376
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1007/s00454-021-00362-9
