Strong uniqueness, Lipschitz continuity, and continuous selections for metric projections in L1

Lower Semicontinuity Concepts, Continuous Selections, and Set Valued Metric Projections

A number of semicontinuity concepts and the relations between them are discussed. Characterizations are given for when the (set-valued) metric projection P M onto a proximinal subspace M of a normed linear space X is approximate lower semicontinuous or 2-lower semicontinuous. A geometric characterization is given of those normed linear spaces X such that the metric projection onto every one-dim...

Lipschitz Continuity of inf-Projections

It is shown that local epi-sub-Lipschitz continuity of the function-valued mapping associated with a perturbed optimization problem yields the local Lipschitz continuity of the inf-projections (= marginal functions, = infimal functions). The use of the theorem is illustrated by considering perturbed nonlinear optimization problems with linear constraints.

Some New Continuity Concepts for Metric Projections

Continuous Selections and Approximations in Α-convex Metric Spaces

In the paper, the notion of a generalized convexity was defined and studied from the view-point of the selection and approximation theory of set-valued maps. We study the simultaneous existence of continuous relative selections and graph-approximations of lower semicontinuous and upper semicontinuous set-valued maps with α-convex values having nonempty intersection.

Semi-continuity of Metric Projections in ∞-direct Sums

Let Y be a proximinal subspace of finite codimension of c0. We show that Y is proximinal in ∞ and the metric projection from ∞ onto Y is Hausdorff metric continuous. In particular, this implies that the metric projection from ∞ onto Y is both lower Hausdorff semi-continuous and upper Hausdorff semi-continuous.