A Characterization of (Locally) Uniformly Convex Spaces in Terms of Relative Openness of Quotient Maps on the Unit Ball

Relative openness of quotient maps on the closed unit ball U of a normed linear space X is studied quantitatively. Particularly, it follows from the results that the quotient maps on X associated with the closed linear subspaces of X are equally relatively open on U if and only if X is locally uniformly convex. Also, X is locally uniformly convex if and only if for any family of linear maps defined on X, equal relative openness on X implies equal relative openness on U. Similarly, uniformly convex spaces can be characterized in terms of equal uniform relative openness of quotient maps on U.