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 $[3d-3, 4d-4]$, and that for a polytope having a line segment for a summand, $4d-4$ is sharp. As an application, the exact lower bound of the number of $k$-faces of a decomposable $d$-polytope with $2d+m$ vertices ($2 \le m\le d-4$) is obtained. Concerning the facets, in dimension 4, the minimum number of facets of a conditionally decomposable polytope is 9, and in dimension $d\ge 5$, the minimum is $d+4$.