Self Jung constants and product spaces.

Let X be a real Banach space and let F be a subset of X which contains at least two points. Let (F) := sup{kx−yk: x, y 2 F} and r(F) := infx2co(F) supy2F kx−yk, where co(F) means the convex hull of F. With this notation it is possible to define the finite self-Jung constant as follows: J(X) := 2 sup{r(F) (F) : F is a finite subset of X}. On the other hand, given two Banach spaces X, Y, consider the product space Z = X Y, with respect to some monotone norm. The main result of this paper is: for every 2 [0, 1], J(Z) 2[(J(X) 2 + (1−))(+(1−)J(Y ) 2 )]. As a consequence of this theorem the authors obtain the following results: 1. J(X 1 Y ) 4−J(X)J(Y ) 4−(J(X)+J(Y )) . 2. If max{J(X), J(Y )} < 2 then J(X 1 Y ) < 2. 3. If = (2−J(X) 2−J(Y ) ) 1 p−1 , 1 p + 1 q = 1, 1 J(X) J(Y ) < 2, J(X) < 2 and 1 < p <1then J(X p Y ) 4−J(X)J(Y ) [(2−J(X))q +(2−J(Y ))q]1 q. By using the above results, the we obtain some fixed point theorems for the product of two Banach spaces via uniformly normal structure. We concludes with a direct proof of the fact that J(X) < 2 implies that X is a B-convex Banach space.