Characterizations of $L^1$-predual spaces by centerable subsets

summary:In this note, we prove that a real or complex Banach space $X$ is an $L^1$-predual space if and only if every four-point subset of $X$ is centerable. The real case sharpens Rao's result in [{\it Chebyshev centers and centerable sets\/}, Proc. Amer. Math. Soc. {\bf 130} (2002), no. 9, 2593--2598] and the complex case is closely related to the characterizations of $L^1$-predual spaces by Lima [{\it Complex Banach spaces whose duals are $L_1$-spaces\/}, Israel J. Math. {\bf 24} (1976), no. 1, 59--72]