“A. R. Alimov and I. G. Tsar’kov; Geometric theory of approximation” Geometricheskaya teoriya priblizhenii) (Russian). Part I. “Classical notions and constructions in the approximation by sets” (Klassicheskie ponyatiya i kontruktsii priblizheniya mnozhestvami), 346 p, OntoPrint Moscow, 2017, ISBN 978-5-906886-91-0; Part II. “Approximation by classes of sets, further developments of basic questions of the geometric approximation theory” (Priblizhenie klassami mnozhestv, dal’neishee razvitie osnovnykh voprosov geometricheskoi teorii priblizhenya), 350 p, OntoPrint Moscow, 2018, ISBN 978-5-00121-053-5.

Abstract

The book is about the best approximation in normed linear spaces in connection with the geometric properties of the underlying space. For a long time the standard reference in this area was Ivan Singer, Best approximation in normed linear spaces by elements of linear subspaces, Springer 1970 (an updated translation of the Romanian version from 1967, see also, I. Singer, The theory of best approximation and functional analysis, SIAM, Philadelphia, PA, 1974). Singer’s books stimulated the research in this area and, since ten, a lot of results were obtained, new notions emerged and many challenging problems were solved. But one, considered by some researchers the most important in best approximation theory, resisted to all attempts to solve it – the problem of the convexity of Chebyshev sets – is any Chebyshev subset of a Hilbert space convex? The authors of this book have important contributions to this problem, mainly concerning the class of the so-called solar sets (or suns), a recurrent theme of the book and an essential tool in the characterization of best approximation (e.g. Kolmogorov’s criterium).