On super fixed point property and super weak compactness of convex subsets in Banach spaces

For a nonempty convex set C of a Banach space X, a self-mapping on C is said to a linear (respectively, affine) isometry if it is the restriction of a linear (respectively, affine) isometry defined on the whole space X. By means of super weakly compact set theory established in the recent years, in this paper, we first show that a nonempty closed bounded convex set of a Banach space has super fixed point property for affine (or, equivalently, linear) isometries if and only if it is super weakly compact; and the super fixed point property and the super weak compactness coincide on every closed bounded convex subset of a Banach space under equivalent renorming sense. With the application of Fabian-Montesinos-Zizler's renorming theorem, we finally show that every strongly super weakly compact generated Banach space can be renormed so that every weakly compact convex set has super fixed point property.