TY - JOUR
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.
UR - http://www.scopus.com/inward/record.url?scp=85016174015&partnerID=8YFLogxK
U2 - 10.1016/S0195-6698(85)80029-9
DO - 10.1016/S0195-6698(85)80029-9
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AN - SCOPUS:85016174015
SN - 0195-6698
VL - 6
SP - 211
EP - 214
JO - European Journal of Combinatorics
JF - European Journal of Combinatorics
IS - 3
ER -