A Simple Proof of the Upper Bound Theorem

N. Alon; G. Kalai

doi:10.1016/S0195-6698(85)80029-9

A Simple Proof of the Upper Bound Theorem

N. Alon, G. Kalai

Research output: Contribution to journal › Article › peer-review

27 Scopus citations

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)
Pages (from-to)	211-214
Number of pages	4
Journal	European Journal of Combinatorics
Volume	6
Issue number	3
DOIs	https://doi.org/10.1016/S0195-6698(85)80029-9
State	Published - 1985
Externally published	Yes

All Science Journal Classification (ASJC) codes

Discrete Mathematics and Combinatorics

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10.1016/S0195-6698(85)80029-9

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title = "A Simple Proof of the Upper Bound Theorem",

abstract = "Let ci(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 ci(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.",

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T1 - A Simple Proof of the Upper Bound Theorem

AU - Alon, N.

AU - Kalai, G.

PY - 1985

Y1 - 1985

N2 - Let ci(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 ci(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.

AB - Let ci(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 ci(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.

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U2 - 10.1016/S0195-6698(85)80029-9

DO - 10.1016/S0195-6698(85)80029-9

M3 - Article

AN - SCOPUS:85016174015

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VL - 6

SP - 211

EP - 214

JO - European Journal of Combinatorics

JF - European Journal of Combinatorics

IS - 3

ER -

A Simple Proof of the Upper Bound Theorem

Abstract

All Science Journal Classification (ASJC) codes

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