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, 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.
