We study -vertex -dimensional polytopes with at most one nonsimplex facet with, say, vertices, called almost simplicial polytopes. We provide tight lower and upper bound theorems for these polytopes as functions of, and, thus generalizing the classical Lower Bound Theorem by Barnette and the Upper Bound Theorem by McMullen, which treat the case where s = 0. We characterize the minimizers and provide examples of maximizers for any. Our construction of maximizers is a generalization of cyclic polytopes, based on a suitable variation of the moment curve, and is of independent interest.
Bibliographical noteFunding Information:
Received by the editors February 4, 2018; revised November 1, 2018. Published online on Cambridge Core May 21, 2019. Research of E. Nevo was partially supported by Israel Science Foundation grants ISF-805/11 and ISF-1695/15. Research of J. Ugon was supported by ARC discovery project DP180100602. AMS subject classification: 52B05, 52B12, 52B22. Keywords: polytope, simplicial polytope, almost simplicial polytope, Lower Bound theorem, Upper Bound theorem, graph rigidity, h-vector, f -vector.
© 2018 Canadian Mathematical Society.
- Lower Bound theorem
- Upper Bound theorem
- almost simplicial polytope
- graph rigidity
- simplicial polytope