Abstract: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 [{Chebyshev centers and centerable sets}, Proc. Amer. Math. Soc. {130} (2002), no. 9, 2593--2598] and the complex case is closely related to the characterizations of $L^1$-predual spaces by Lima [{Complex Banach spaces whose duals are $L_1$-spaces}, Israel J. Math. {24} (1976), no. 1, 59--72].
Keywords: Chebyshev radius, centerable subsets and $L^1 $-predual spaces
AMS Subject Classification: Primary 41A65; Secondary 46B20