### Abstract

Let c_{i}(n, d) be the number of i-dimensional faces of a cyclic d-polytope on n vertices. We present a simple new proof of the upper bound theorem for convex polytopes, which asserts that the number of i-dimensional faces of any d-polytope on n vertices is at most c_{i}(n, d). Our proof applies for arbitrary shellable triangulations of (d−1) spheres. Our method provides also a simple proof of the upper bound theorem for d-representable complexes.

Original language | English (US) |
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Pages (from-to) | 211-214 |

Number of pages | 4 |

Journal | European Journal of Combinatorics |

Volume | 6 |

Issue number | 3 |

DOIs | |

State | Published - Jan 1 1985 |

Externally published | Yes |

### All Science Journal Classification (ASJC) codes

- Discrete Mathematics and Combinatorics

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## Cite this

Alon, N., & Kalai, G. (1985). A Simple Proof of the Upper Bound Theorem.

*European Journal of Combinatorics*,*6*(3), 211-214. https://doi.org/10.1016/S0195-6698(85)80029-9