Vol. 46, No. 2, 1973

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A proof of the lower bound conjecture for convex polytopes

David Wilmot Barnette

Vol. 46 (1973), No. 2, 349–354

A d-polytope is defined to be a d-dimensional set that is the convex hull of a finite number of points. A d-polytope is said to be simplicial if each facet is a simplex. Dually, a d-polytope is simple if each vertex has valence d. It has been conjectured that the following inequalities hold for any simplicial d-polytope p.

fk (d )
kf0 (d+ 1)
k +1k for all 1 k d 2 (1)
fd1 (d 1)f0 (d + 1)(d 2) (2)
Here, fi is the number of i-dimensional faces of P. This conjecture is known as the Lower Bound Conjecture, hereafter to be abbreviated LBC. The LBC has been known to be true for d 3 for quite some time. In 1969, D. Walkup proved the LBC for d = 4 and 5. In 1970, the author proved (2) for all simplicial d-polytopes. In this paper (1) is proved for all simplicial d-polytopes.

Mathematical Subject Classification
Primary: 52A25
Received: 6 April 1972
Published: 1 June 1973
David Wilmot Barnette