Geodesic
Almost Simplicial Polytopes: The Lower and Upper Bound Theorems
Canadian journal of mathematics, Tome 72 (2020) no. 2, pp. 537-556

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DOI

We study $n$-vertex $d$-dimensional polytopes with at most one nonsimplex facet with, say, $d+s$ vertices, called almost simplicial polytopes. We provide tight lower and upper bound theorems for these polytopes as functions of $d,n$, and $s$, 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 $d$. Our construction of maximizers is a generalization of cyclic polytopes, based on a suitable variation of the moment curve, and is of independent interest.
DOI : 10.4153/S0008414X18000123
Mots-clés : polytope, simplicial polytope, almost simplicial polytope, Lower Bound theorem, Upper Bound theorem, graph rigidity, h-vector, f-vector 
Nevo, Eran; Pineda-Villavicencio, Guillermo; Ugon, Julien; Yost, David. Almost Simplicial Polytopes: The Lower and Upper Bound Theorems. Canadian journal of mathematics, Tome 72 (2020) no. 2, pp. 537-556. doi: 10.4153/S0008414X18000123
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     title = {Almost {Simplicial} {Polytopes:} {The} {Lower} and {Upper} {Bound} {Theorems}},
     journal = {Canadian journal of mathematics},
     pages = {537--556},
     year = {2020},
     volume = {72},
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     doi = {10.4153/S0008414X18000123},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/S0008414X18000123/}
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