  The conjecture of Kalai, Kleinschmidt, and Lee on the number of empty
simplices of a simplicial polytope is established by relating it to the first
graded Betti numbers of the polytope. The proof allows us to derive explicit
optimal bounds on the number of empty simplices of any given dimension. As a
key result, we prove optimal bounds for the graded Betti numbers of any
standard graded K-algebra in terms of its Hilbert function.
