Let B := (B, || · ||b) be a Banach space and H := (H, || · ||h) a dense subspace of B. We are interested in the rate at which the distance of a given x∈ B from the ball of radius t in H tends to zero as t →1. This problem has numerous applications and its recent importance in the Cucker Smale theory of learning renewed our interest in this subject. We study two aspects of this problem. First, we obtain general results on the relation between the CS-functional and the Peetre K-functional. Secondly, we estimate it in the concrete case of Besov spaces by using functions of exponential type.