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# Polytopes and $C^1$-convex bodies

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 $\{P_n\}_{n=0}^{\infty}$ 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.
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https://hal.inria.fr/hal-01207607
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Submitted on : Thursday, October 1, 2015 - 9:29:11 AM
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Distributed under a Creative Commons Attribution 4.0 International License

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• HAL Id : hal-01207607, version 1

### Citation

Karim Adiprasito, José Alejandro Samper. Polytopes and $C^1$-convex bodies. 26th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2014), 2014, Chicago, United States. pp.277-288. ⟨hal-01207607⟩

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