Lipschitz retraction of finite subsets of Hilbert spaces (Q2786587)

In this article the author continues his studies of the hyperspaces \(X(n)\) of all subsets of size at most \(n\) of a given metric space \((X,d)\), equipped with the Hausdorff distance induced by \(d\). He proves that for each \(n > 1\), there exists a Lipschitz retraction from \(H(n)\) onto \(H(n-1)\) when \(H\) is an arbitrary Hilbert space. As a consequence, it is shown that the space \(\mathbb{R}^d(n)\) is an absolute Lipschitz retract.NEWLINENEWLINEThe paper is concluded with some open questions dealing with Lipschitz retractions. One of them asks whether it is possible to construct retractions from \(H(n)\) onto \(H(n-1)\) (where \(H\) is a Hilbert space) with a Lipschitz constant independent of \(n\); and the other one asks whether in the first result cited above the Hilbert space can be replaced by any Banach space.