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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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Convex Union Representability and Convex Codes
Abstract We introduce and investigate $$d$$-convex union representable complexes: the simplicial complexes that arise as the nerve of a finite collection of convex open sets in $${\mathbb{R}}^d$$ whose union is also convex. Chen, Frick, and Shiu recently proved that such complexes are collapsible and asked if all collapsible complexes are convex union representable. We disprove this by showing that there exist shellable and collapsible complexes that are not convex union representable; there also exist non-evasive complexes that are not convex union representable. In the process we establish several necessary conditions for a complex to be convex union representable such as that such a complex $$\Delta $$ collapses onto the star of any face of $$\Delta $$, that the Alexander dual of $$\Delta $$ must also be collapsible, and that if $$k$$ facets of $$\Delta $$ contain all free faces of $$\Delta $$, then $$\Delta $$ is $(k-1)$-representable. We also discuss some sufficient conditions for a complex to be convex union representable. The notion of convex union representability is intimately related to the study of convex neural codes. In particular, our results provide new families of examples of non-convex neural codes.
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- Award ID(s):
- 1664865
- PAR ID:
- 10281750
- Date Published:
- Journal Name:
- International Mathematics Research Notices
- Volume:
- 2021
- Issue:
- 9
- ISSN:
- 1073-7928
- Page Range / eLocation ID:
- 7132 to 7158
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
-
Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1093/imrn/rnz055
