On the reconstruction of polytopes

Blind and Mani, and later Kalai, showed that the face lattice of a simple polytope is determined by its graph, namely its 1-skeleton. Call a vertex of a d-polytope nonsimple if the number of edges incident to it is more than d. We show that (1) the face lattice of any d-polytope with at most two nonsimple vertices is determined by its 1-skeleton; (2) the face lattice of any d-polytope with at most (Formula presented.) nonsimple vertices is determined by its 2-skeleton; and (3) for any (Formula presented.) there are two d-polytopes with (Formula presented.) nonsimple vertices, isomorphic (Formula presented.)-skeleta and nonisomorphic face lattices. In particular, the result (1) is best possible for 4-polytopes.