## Abstract

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 {Pn}^{8}_{n}=0 of simplicial polytopes converges to a C^{1}-convex body in the Hausdorff distance, then the entries of the g-vector of Pn converge to infinity.

Original language | American English |
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Pages (from-to) | 277-287 |

Number of pages | 11 |

Journal | Discrete Mathematics and Theoretical Computer Science |

State | Published - 2014 |

Externally published | Yes |

Event | 26th International Conference on Formal Power Series and Algebraic Combinatorics, FPSAC 2014 - Chicago, United States Duration: 29 Jun 2014 → 3 Jul 2014 |

### Bibliographical note

Funding Information:∗Email: adiprasito@ihes.fr. K. Adiprasito has been supported by an EPDI postdoctoral fellowship and by the Romanian NASR, CNCS — UEFISCDI, project PN-II-ID-PCE-2011-3-0533. †Email: samper@math.washington.edu. Jose Samper’s research is partially supported by a graduate fellowship from NSF Grant DMS-1069298

Publisher Copyright:

© 2014 Discrete Mathematics and Theoretical Computer Science (DMTCS), Nancy, France

## Keywords

- Approximation theory
- Convex bodies
- F-vector theory
- Geometric Combinatorics
- Lower bound theorem
- Polytopes

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