On polytopes simple in edges

In this paper, we will study the h-vectors of a slightly more general class of polytopes. A d-polytope is called simple in edges if each its edge is incident exactly to d − 1 facets. We will prove that for any polytope simple in edges all numbers h[d/2], h[d/2]+1,. . ., hd are nonnegative and hk 6 hd−k for k 6 d/2. Polytopes simple in edges appear for instance as (closures of) fundamental polyhedra of groups generated by reflections in Lobachevskii spaces. A combinatorial study of polytopes simple in edges carried out by Khovanskii [4] concluded the proof