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.

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.

Reviewer: Stefan Cobzaş (Cluj-Napoca)

##### MSC:

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) |