Geodesic
Almost simplicial polytopes: the lower and upper bound theorems
Discrete mathematics & theoretical computer science, DMTCS Proceedings, 28th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2016), DMTCS Proceedings, 28th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2016) (2020).

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this is an extended abstract of the full version. 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 bounds for the face numbers of these polytopes as functions of d, n and s, thus generalizing the classical Lower Bound Theorem by Barnette and Upper Bound Theorem by McMullen, which treat the case s = 0. We characterize the minimizers and provide examples of maximizers, for any d. 
@article{DMTCS_2020_special_379_a51,
     author = {Nevo, Eran and Pineda-Villavicencio, Guillermo and Ugon, Julien and Yost, David},
     title = {Almost simplicial polytopes: the lower and upper bound theorems},
     journal = {Discrete mathematics & theoretical computer science},
     publisher = {mathdoc},
     volume = {DMTCS Proceedings, 28th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2016)},
     year = {2020},
     doi = {10.46298/dmtcs.6369},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.46298/dmtcs.6369/}
}
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%V DMTCS Proceedings, 28th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2016)
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Nevo, Eran; Pineda-Villavicencio, Guillermo; Ugon, Julien; Yost, David. Almost simplicial polytopes: the lower and upper bound theorems. Discrete mathematics & theoretical computer science, DMTCS Proceedings, 28th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2016), DMTCS Proceedings, 28th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2016) (2020). doi : 10.46298/dmtcs.6369. http://geodesic.mathdoc.fr/articles/10.46298/dmtcs.6369/

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