NECESSARY AND SUFFICIENT CONVEXITY CONDITIONS FOR THE RANGES OF VECTOR MEASURES

The absence of atoms in Lyapunov's Convexity Theorem is a sufficient, but not a necessary condition for the convexity of the range of an n-d imensional vector measure. In this paper algebraic and topological con vexity conditions generalizing Lyapunov's Theorem are developed which are sufficient and necessary as well. From these results the converse of Lyapunov's Theorem is derived in the form of a nonconvexity stateme nt which gives insight into the geometric structure of the ranges of v ector measures with atoms. Further, a characterization of the one-dime nsional faces of a zonoid Z(mu) is given with respect to the generatin g spherical Borel measure mu. As an application, it is shown that the absence of mu-atoms is a necessary and sufficient convexity condition for the range of the indefinite integral integral x d mu, where x deno tes the identical function on S-n-1.