A proof of the lower bound conjecture for convex polytopes

David Wilmot Barnette

Vol. 46 (1973), No. 2, 349–354

DOI: 10.2140/pjm.1973.46.349

Abstract

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	≧f0 −k for all 1 ≦ k ≦ d − 2	(1)
fd−1	≧ (d − 1)f0 − (d + 1)(d − 2)	(2)

Here, f_i 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