Lower bound results for conditionally decomposable polytopes

It is possible for a combinatorial type of polytope to have both decomposable and indecomposable realizations; here decomposability is meant with respect to Minkowski addition. Such polytopes are called conditionally decomposable. We show that the minimum number of vertices of a conditionally decomposable d -polytope is in the range [3 d − 3, 4 d − 4], and that for a polytope having a line segment for a summand, 4 d − 4 is sharp. As an application, the exact lower bound of the number of k -faces of a decomposable d -polytope with 2 d + m (1 ≤ m ≤ d − 4) vertices is obtained. Concerning the facets, in dimension 4 the minimum number of facets of a conditionally decomposable polytope is 9, and in dimension d ≥ 5 the minimum is d + 4.