We call a Banach space X admitting the Mazur-Ulam property (MUP) provided that for any Banach space Y , if f is an onto isometry between the two unit spheres of X and Y , then it is the restriction of a linear isometry between the two spaces. A generalized MazurUlam question is whether every Banach space admits the MUP. In this paper, we show first that the question has an affirmative answer for a general class of Banach spaces, namely, somewhere-flat spaces. As their immediate consequences, we obtain on the one hand that the question has an approximately positive answer: Given ε > 0, every Banach space X admits a (1 + ε)-equivalent norm such that X has the MUP; on the other hand, polyhedral spaces, CL-spaces admitting a smooth point (in particular, separable CL-spaces) have the MUP.

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This paper mainly concerns the topological nature of uniformly convexifiable sets in general Banach spaces: A sufficient and necessary condition for a bounded closed convex set C of a Banach space X to be uniformly convexifiable (i.e. there exists an equivalent norm on X which is uniformly convex on C) is that the set C is super-weakly compact, which is defined using a generalization of finite representability. The proofs use appropriate versions of classical theorems, such as James' finite tree theorem, Enflo's renorming technique, Grothendieck's lemma and the Davis-Figiel-Johnson-Pełczyński lemma.

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The question of establishing measure theory for statistical convergence has been moving closer to center stage, since a kind of reasonable theory is not only fundamental for unifying various kinds of statistical convergence, but also a bridge linking the studies of statistical convergence across measure theory, integration theory, probability and statistics. For this reason, this paper, in terms of subdifferential, first shows a representation theorem for all finitely additive probability measures defined on the σ-algebra A of all subsets of N , and proves that every such measure can be uniquely decomposed into a convex combination of a countably additive probability measure and a statistical measure (i.e. a finitely additive probability measure μ with μ(k) = 0 for all singletons {k}). This paper also shows that classical statistical measures have many nice properties, such as: The set S of all such measures endowed with the topology of point-wise convergence on A forms a compact convex Hausdorff space; every classical statistical measure is of continuity type (hence, atomless), and every specific class of statistical measures fits a complementation minimax rule for every subset in N . Finally, this paper shows that every kind of statistical convergence can be unified in convergence of statistical measures.