On the generalized lower bound conjecture for polytopes and spheres

In 1971, McMullen and Walkup posed the following conjecture, which is called the generalized lower bound conjecture: If P is a simplicial d -polytope then its h -vector ( h 0 , h 1 , …, h d ) satisfies . Moreover, if h r −1 = h r for some then P can be triangulated without introducing simplices of dimension ≤ d − r . The first part of the conjecture was solved by Stanley in 1980 using the hard Lefschetz theorem for projective toric varieties. In this paper, we give a proof of the remaining part of the conjecture. In addition, we generalize this result to a certain class of simplicial spheres, namely those admitting the weak Lefschetz property.