Some remarks on measures of noncompactness and retractions onto spheres

Optimal retraction problem for proper $k$-ball-contractive mappings in $C^m [0,1]$

“…The fact of considering another metric property, namely measures of noncompactness of the above retractions leads to more useful results for applications as, for instance, applications to theorems of Birkhoff-Kellog type (see [8,9,11,16,21]). Let us recall that the Hausdorff measure of noncompactness γ(A) of a bounded subset A of X is the infimum of all ε > 0 such that A has a finite ε-net in X.…”

An application of approach theory to the relative Hausdorff measure of non-compactness for the Wasserstein metric

“…Instead of quantifying the properties of a space X by means of one metric, an approach space is determined by assigning to each point x ∈ X a collection A x of [0, ∞]-valued maps on X which are interpreted as local distances based at x. By imposing suitable axioms on the collections A x , we obtain a structure on X which, as in the case of metric spaces, allows us to deal with quantitative concepts such as asymptotic radius and center ( [AMS82], [B85], [L81], [L01]) and Hausdorff measure of non-compactness ( [BG80], [L88]) as used in functional analysis, especially in various areas of approximation theory ( [AMS82]), fixed-point theory ( [GK90]), operator theory ( [AKP92], [PS88]), and Banach space geometry ( [DB86], [KV07], [WW96]). However, unlike metric spaces, approach spaces share many of the 'structurally good' properties of topological spaces, such as e.g.…”