Geodesic
Polytopes and $C^1$-convex bodies
Discrete mathematics & theoretical computer science, DMTCS Proceedings vol. AT, 26th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2014), DMTCS Proceedings vol. AT, 26th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2014) (2014).

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The face numbers of simplicial polytopes that approximate $C^1$-convex bodies in the Hausdorff metric is studied. Several structural results about the skeleta of such polytopes are studied and used to derive a lower bound theorem for this class of polytopes. This partially resolves a conjecture made by Kalai in 1994: if a sequence $\{P_n\}_{n=0}^{\infty}$ of simplicial polytopes converges to a $C^1$-convex body in the Hausdorff distance, then the entries of the $g$-vector of $P_n$ converge to infinity. 
@article{DMTCS_2014_special_265_a24,
     author = {Adiprasito, Karim and Samper, Jos\'e Alejandro},
     title = {Polytopes and $C^1$-convex bodies},
     journal = {Discrete mathematics & theoretical computer science},
     publisher = {mathdoc},
     volume = {DMTCS Proceedings vol. AT, 26th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2014)},
     year = {2014},
     doi = {10.46298/dmtcs.2399},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.46298/dmtcs.2399/}
}
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Adiprasito, Karim; Samper, José Alejandro. Polytopes and $C^1$-convex bodies. Discrete mathematics & theoretical computer science, DMTCS Proceedings vol. AT, 26th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2014), DMTCS Proceedings vol. AT, 26th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2014) (2014). doi : 10.46298/dmtcs.2399. http://geodesic.mathdoc.fr/articles/10.46298/dmtcs.2399/

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