Chebyshev sets vs convex sets: the situation from the perspective of convex nonsmooth analysis. (Ensembles de Tchebychev vs. ensembles convexes: l’état de la situation vu via l’analyse convexe non lisse.) (French) Zbl 1098.49506
Summary: A closed set \(S\) of a Hilbert space \(H\) is said to be Chebyshev if every point in \(H\) has one and only one closest point in \(S\). Is such a set necessarily convex? This question, one of the most famous in Approximation and Real Analysis, has not been completely answered as yet, at least when \(H\) is infinite dimensional. The aim of the present work is precisely to take stock of this question, emphasizing the viewpoint of Convex Nonsmooth Analysis.
MSC:
| 49J52 | Nonsmooth analysis |
| 41A65 | Abstract approximation theory (approximation in normed linear spaces and other abstract spaces) |
