In this paper we characterize Property-(R1 ), a generalization of 11 ball property. As a necessary and sufficient condition of a subspace Y 2 with Property-(R1 ) we derive that r(y, F) = radY (F ) + d(y, centY (F )) for any bounded subset F and y ∈ Y . We introduce the notion of modulus of relative chebyshev centre and characterize Property-(R1 ) in terms of this modulus. It is observed that if Y is a finite co-dimensional strongly proximinal subspace of a L1 predual space X and F is a finite subset of X then radY (F ) = radX (F ) + d(F, Y ). We characterize continuity of centV (.) in terms of the modulus of relative chebyshev centre. © 2019, American Mathematical Society. All rights reserved.