On Polytopes that are Simple at the Edges

We study some combinatorial properties of polytopes that are simple at the edges. We give an elementary geometric proof of an analog of the hard Lefschetz theorem for the polytopes for which the distance between any two nonsimple vertices is sufficiently large. This implies that the $h$-vector of such polytopes satisfies the relations $h_{[d/2]}\ge h_{[d/2]+1}\ge\cdots\ge h_d=1$, where $d$ is the dimension of the polytope, which proves a special case of Stanley's conjecture.