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On the convexity of \( N\)-Chebyshev sets. (English. Russian original) Zbl 1239.46013
Izv. Math. 75, No. 5, 889-914 (2011); translation from Izv. Ross. Akad. Nauk, Ser. Mat. 75, No. 5, 19-46 (2011).
For a nonempty subset \(M\) of a normed space \((X,\|\cdot\|)\), a natural number \(N\geq 1\) and \(x_1,\dots,x_N\in X\) put \(\rho(x_1,\dots,x_N;M)=\inf\{\|y-x_1\|+\dots+\|y-x_N\| : y\in M\} \) and \(P_M(x_1,\dots,x_N)=\{y\in M : \|y-x_1\|+\dots+\|y-x_N\|= \rho(x_1,\dots,x_N;M)\}.\) Let also \(\rho(x_1,\dots,x_N)=\rho(x_1,\dots,x_N;X).\) The set \(M\) is called \(N\)-Chebyshev if for all \(N\)-tuples \( x_1,\dots,x_N\in X,\,\) (i) \(\#P_M(x_1,\dots,x_N)=1\) if \(\rho(x_1,\dots,x_N;M)>\rho(x_1,\dots,x_N)\) and (ii) \(\#P_M(x_1,\dots,x_N)\geq 1\) if \(\rho(x_1,\dots,x_N;M)=\rho(x_1,\dots,x_N)\, \) (the condition (ii) ensures that an \(N\)-Chebyshev set is closed). It is clear that a 1-Chebyshev set is Chebyshev in the usual sense. Every \(N\)-Chebyshev set is \(d\)-Chebyshev for every natural divisor \(d\) of \(N\), so that it is always Chebyshev (the case \(d=1\)). The author gives an example of a Chebyshev set which is not 2-Chebyshev.
It is known that any closed convex subset of a reflexive strictly convex Banach space \(X\) is Chebyshev. The author shows that a similar result holds for \(N\)-Chebyshev sets. If further, the space \(X\) is finite dimensional, then every \(N\)-Chebyshev subset of \(X\) is closed and convex for any even \(N\) (Corollary 3).
A famous unsolved problem in best approximation theory is the following: Must any Chebyshev subset of a Hilbert space be convex? For a survey of results obtained in the study of this problem (up to 1996) see the paper by V. S. Balaganskij and L. P. Vlasov [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; Zbl 0931.41017)]. In a previous paper [Vestn. Mosk. Univ., Ser. I 2008, No. 3, 16–19 (2008); translation in Mosc. Univ. Math. Bull. 63, No. 3, 96–98 (2008; Zbl 1212.52002)], the author proved that any 2-Chebyshev subset of a Hilbert space is closed and convex. The present paper is concerned with the convexity of \(N\)-Chebyshev subsets of Banach spaces. If \(N\) is even, then any \(N\)-Chebyshev subset of a uniformly convex Banach space \(X\) is closed and convex; if further, the space \(X\) is also smooth, then the result is true for all odd \(N\geq 3,\,\) too (Theorem 3). In Section 3 the problem of convexity of 2-Chebyshev sets is studied within a class of Banach spaces, called spaces with narrow 2-balls, which contains the class of strictly convex Banach spaces and is contained (unknown whether strictly) in the class of locally uniformly convex ones. A 2-ball is a set of the form \(B(x_1,x_2;r)=\{y\in X : \|y-x_1\|+\|y-x_2\| \leq r\},\) for \(x_1,x_2\in X\) and \(r>0.\)
In Section 5, some results of L. P. Vlasov [Mat. Zametki 3, 59–69 (1968; Zbl 0155.45401)]; translation in [Math. Notes 3, 36–41 (1968; Zbl 0164.15004)] on the connectedness properties of the metric projection, are extended to uniformly convex asymmetrically normed spaces. In the last section, Section 7, one proves that in an arbitrary Banach space, a subset which is \(N\)-Chebyshev for infinitely many \(N\) is closed and convex (Theorem 6). The paper contains also other related results and examples.

46B20 Geometry and structure of normed linear spaces
41A50 Best approximation, Chebyshev systems
41A52 Uniqueness of best approximation
41A65 Abstract approximation theory (approximation in normed linear spaces and other abstract spaces)
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