For a closed subspace Y of a Banach space X, we define a separably determined property for Y in X. We prove that if the strong 11/ 2-ball property is separably determined for Y in X, then L 1(μ, Y) has the strong 11/2ball property in L 1(μ, X). For an M-embedded space X, we give a class of elements in L 1(μ, X having best approximations from L 1(μ, X). We also prove that some of the proximinality properties are stable under polyhedral direct sums of Banach spaces. © 2014 Copyright Taylor & Francis Group, LLC.