Uniform Convexity and Convergence of a Sequence of Sets in a Complete Geodesic Space

In this paper, we first introduce two new notions of uniform convexity on a geodesic space, and we prove their properties. Moreover, we reintroduce a concept of the set-convergence in complete geodesic spaces, and we prove a relation between the metric projections and the convergence of a sequence of sets.

The Representations and Continuity of the Metric Projections on Two Classes of Half-Spaces in Banach Spaces

We show a necessary and sufficient condition for the existence of metric projection on a class of half-spaceKx0*,c={x∈X:x*(x)≤c}in Banach space. Two representations of metric projectionsPKx0*,candPKx0,care given, respectively, whereKx0,cstands for dual half-space ofKx0*,cin dual spaceX*. By these representations, a series of continuity results of the metric projectionsPKx0*,candPKx0,care given. We also provide the characterization that a metric projection is a linear bounded operator.