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.