Continuity of Metric Projections

Some New Continuity Concepts for Metric Projections

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.

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.

On projections of metric spaces

Let X be a metric space and let μ be a probability measure on it. Consider a Lipschitz map T : X → Rn, with Lipschitz constant ≤ 1. Then one can ask whether the image TX can have large projections on many directions. For a large class of spaces X, we show that there are directions φ ∈ Sn−1 on which the projection of the image TX is small on the average (in L2(μ)), with bounds depending on the d...

ON THE CONTINUITY OF PROJECTIONS AND A GENERALIZED GRAM-SCHMIDT PROCESS

Let ? be an open connected subset of the complex plane C and let T be a bounded linear operator on a Hilbert space H. For ? in ? let e the orthogonal projection onto the null-space of T-?I . We discuss the necessary and sufficient conditions for the map ?? to b e continuous on ?. A generalized Gram- Schmidt process is also given.