A bounded convex subset K of
a Banach space B has normal structure if for each convex subset H of K which
contains more than one point there is a point x in H which is not a diametral point
of H. The concept of normal structure (introduced by Brodskii and Milman) and a
strengthening of this concept called complete normal structure have been of
fundamental importance in some recent investigations concerned with determining
conditions on weakly compact K under which the members of any commutative
family 𝒯 of nonexpansive mappings of K into itself have a common fixed-point. A
more thorough study of these concepts is initiated in the present paper. The
theorems obtained primarily concern product spaces composed of spaces which
possess normal structure.
