Karim Adiprasito ; José Alejandro Samper

Polytopes and $C^1$convex bodies
dmtcs:2399 
Discrete Mathematics & Theoretical Computer Science,
January 1, 2014,
DMTCS Proceedings vol. AT, 26th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2014)

https://doi.org/10.46298/dmtcs.2399
Polytopes and $C^1$convex bodiesArticle
Authors: Karim Adiprasito ^{1}; José Alejandro Samper ^{2}
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.