On The Number of Faces of a Convex Polytope

The following problem is as yet unsolved: Given a convex polytope with N vertices in n -space, what is the maximum number of ( n — 1)-faces which it can have? Aside from its geometric interest this question arises in connection with solving systems of linear inequalities and linear equations in non-negative variables. The problem is equivalent to asking for the best bound on the number of basic solutions for such problems and hence a bound (though a weak one) for the number of iterations needed in the simplex method for solving linear programmes.