The Lyapunov theorem on convexity and its use for sign-embeddings - Kadets V. М.; Popov М. М. (Ukrains’kyi Matematychnyi Zhurnal, 1992)
It is proved (Theorem 1) that for a Banach space X the following statements are equialent: i) the range of every X-valued σ-additive non-atomic measure of finite variation has convex closure; ii) L_1 does not sign-embed in X.