Continuous Metric Projections

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...

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...

Characterization of, Reflexive Spaces ems of Continuous Approximate Selec for Metric Projections

Let (X, z) be a topological space, and (Y, d) a metric space. A mapping F: X -+ 2’ which associates with every x E X a non-empty subset F(x) OF Y is said to be lower semi-continuous (I.s.c.) (respectively, upper semi-continuous (u.s.c.)) if, for each open set % in Y, the set (X E X: F(x) n %f # @ > (respectively, the set (X E X: F(x) c @ )) is open in X A mapping f: X-a Y is a selection for F i...

Some New Continuity Concepts for Metric Projections

Weighted continuous metric scaling

Weighted metric scaling (Cuadras and Fortiana 1995b) is a natural extension of classic metric scaling which encompasses several well-known techniques of Euclidean representation of data, including correspondence analysis of bivariate contingency tables. A continuous version of this technique, following the trend initiated in (Cuadras and For-tiana 1993, 1995a) is applied to the study of several...