The problem of convexity of Chebyshev sets.

*(English. Russian original)*Zbl 0931.41017
Russ. Math. Surv. 51, No. 6, 1127-1190 (1996); translation from Usp. Mat. Nauk 51, No. 6, 125-188 (1996); errata ibid. 52, No. 1, 237 (1997).

A nonvoid closed subset \(M\) of a normed space \(X\) is called Chebyshev if every point \(x\in X\setminus M\) has exactly one nearest point in \(M.\) It is well-known that a closed convex set of a strictly convex reflexive Banach space (\(B-\)space) is Chebyshev. The converse, known as the problem of the convexity of Chebyshev sets, and asking for the characterization of those Banach spaces in which every Chebyshev set is convex, turned to be a very difficult one. It is still unsolved, even in the case of a Hilbert space: Must a Chebyshev subset of a Hilbert space be convex? As it was shown by N. Kritikos [Bull. Math. Soc. Roum. Sci. 40, No. 1/2 87-92 (1938; Zbl 0020.07701)] (based on some previous results of L. N. H. Bunt and T. S. Motzkin) the answer is positive for the Euclidean space \(\mathbb R^n.\) The characterization of finite dimensional normed spaces in which every Chebyshev set is convex was given by N. V. Efimov and S. B. Stechkin [Dokl. Akad. Nauk SSSR 127, 254-257 (1959; Zbl 0095.08903)] as being the strictly convex and smooth normed spaces. The characterization of finite dimensional normed spaces in which every bounded Chebyshev set is convex was done by I. G. Tsar’kov [Mat. Zametki 40, No. 2, 174-196 (1986; Zbl 0625.46017)] these are the spaces for which the set of extreme points of the unit ball of the conjugate space is dense in the unit sphere of this space. The problem of convexity of Chebyshev sets in infinite dimensional spaces was considered by N. V. Efimov and S. B. Stechkin in a series of papers published between the years 1958 and 1962 in Doklady. They have shown that the set \(R^n _m\) of rational functions (\(m\) the degree of denominator and \(n\) the degree of the numerator) is not Chebyshev in \(L^p\), \(1<p<\infty\) (although it is a nonconvex Chebyshev subset of \(C[0,1]\)), and raised the problem of convexity of Chebyshev sets in the Hilbert space. The same problem was posed by V. Klee [Math. Ann. 142, No. 3, 293-304 (1961; Zbl 0091.27701)], who considered also the problem of the existence in a Hilbert space of a Chebyshev set with convex complement. Later, E. Asplund [Trans. Am. Math. Soc. 144, 235-240 (1969; Zbl 0187.05504)] has shown that the existence of such a set is equivalent to the existence of a nonconvex Chebyshev set. G. G. Johnson [J. Approximation Theory 51, No. 4, 289-332 (1987; Zbl 0637.41033)] gave an example of Chebyshev set with bounded convex complement in the incomplete inner product space of all eventually null sequences in \(l^2\) (a simplified version of Johnson’s construction is presented in the paper under review). V. Klee [Math. Ann. 257, 251-160 (1981; Zbl 0459.41018), and Stud. Sci. Math. Hung. 21, 415-427 (1986; Zbl 0577.52007)], gave an example (included also in the present paper) of a discrete Chebyshev subset of the space \(l^1 (S),\) with \(S\) uncountable. An example of a Chebyshev subset of \( C[0,1] \) with one isolated point was given by C. B. Dunham [Can. Math. Bull. 18, 35-57 (1975; Zbl 0309.41025)].

But the problem is still unsolved in general. L. P. Vlasov, Mat. Zametki 7, No. 5, 593-604 (1970; Zbl 0198.45903)], proved that Chebyshev sets in uniformly convex \( B-\)spaces are connected. The problem attracted many mathematicians, renowned specialists in geometric functional analysis and in abstract approximation theory, including the authors of the present paper. The situation up to 1973 is presented in the survey paper of L. P. Vlasov [Usp. Mat. Nauk 28, No. 6(174), 3-66 (1973; Zbl 0293.41031)], the bibliography given therein counting 140 items. The present paper can be considered as a sequel to Vlasov’s paper and it reports on further 148 references. A survey paper on the same topic was written by T. D. Narang [Nieuw Arch. Wiskd., III. Ser. 25, 377-402 (1977; Zbl 0372.41021)].

The problem has very strong connections with some geometric properties of Banach spaces – smoothness, strict, uniform or local uniform convexity etc. The so called “sun properties” (there are a lot of suns – \(\alpha\)-, \(\beta\)-, \(\gamma\)-, \(\delta\)-suns, metasuns, strong suns) as well as various continuity and differentiability properties of the metric projection are also relevant in the study of convexity of Chebyshev sets. In the last time methods from nonlinear and convex analysis and from the theory of monotone operators were also used.

The paper under review can be considered as a sample of “condensed matter” containing a huge amount of information. The paper is very well organized and clearly written (a little too technical, in the reviewer’s opinion). Numerous diagrams help the reader to navigate through various classes of spaces and sets and to see the connections between them.

The paper is dedicated to the memory of S. B. Stechkin for his deep and outstanding contributions to approximation theory. The authors conclude the paper expressing the hope that the problem will be solved in the next future. (The conclusions of T. D. Narang (loc. cit.) were less enthusiastic, considering, in 1973, the problem as being far from its solution).

But the problem is still unsolved in general. L. P. Vlasov, Mat. Zametki 7, No. 5, 593-604 (1970; Zbl 0198.45903)], proved that Chebyshev sets in uniformly convex \( B-\)spaces are connected. The problem attracted many mathematicians, renowned specialists in geometric functional analysis and in abstract approximation theory, including the authors of the present paper. The situation up to 1973 is presented in the survey paper of L. P. Vlasov [Usp. Mat. Nauk 28, No. 6(174), 3-66 (1973; Zbl 0293.41031)], the bibliography given therein counting 140 items. The present paper can be considered as a sequel to Vlasov’s paper and it reports on further 148 references. A survey paper on the same topic was written by T. D. Narang [Nieuw Arch. Wiskd., III. Ser. 25, 377-402 (1977; Zbl 0372.41021)].

The problem has very strong connections with some geometric properties of Banach spaces – smoothness, strict, uniform or local uniform convexity etc. The so called “sun properties” (there are a lot of suns – \(\alpha\)-, \(\beta\)-, \(\gamma\)-, \(\delta\)-suns, metasuns, strong suns) as well as various continuity and differentiability properties of the metric projection are also relevant in the study of convexity of Chebyshev sets. In the last time methods from nonlinear and convex analysis and from the theory of monotone operators were also used.

The paper under review can be considered as a sample of “condensed matter” containing a huge amount of information. The paper is very well organized and clearly written (a little too technical, in the reviewer’s opinion). Numerous diagrams help the reader to navigate through various classes of spaces and sets and to see the connections between them.

The paper is dedicated to the memory of S. B. Stechkin for his deep and outstanding contributions to approximation theory. The authors conclude the paper expressing the hope that the problem will be solved in the next future. (The conclusions of T. D. Narang (loc. cit.) were less enthusiastic, considering, in 1973, the problem as being far from its solution).

Reviewer: S.Cobzaş (Cluj-Napoca)