TY - JOUR
T1 - Almost Simplicial Polytopes
T2 - The Lower and Upper Bound Theorems
AU - Nevo, Eran
AU - Pineda-Villavicencio, Guillermo
AU - Ugon, Julien
AU - Yost, David
N1 - Funding 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. Publisher Copyright: © 2018 Canadian Mathematical Society.
PY - 2020/4/1
Y1 - 2020/4/1
N2 - 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.
AB - 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.
KW - Lower Bound theorem
KW - Upper Bound theorem
KW - almost simplicial polytope
KW - f-vector
KW - graph rigidity
KW - h-vector
KW - polytope
KW - simplicial polytope
UR - http://www.scopus.com/inward/record.url?scp=85083063019&partnerID=8YFLogxK
U2 - https://doi.org/10.4153/S0008414X18000123
DO - https://doi.org/10.4153/S0008414X18000123
M3 - Article
SN - 0008-414X
VL - 72
SP - 537
EP - 556
JO - Canadian Journal of Mathematics
JF - Canadian Journal of Mathematics
IS - 2
ER -