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 $X$.