List of Alexey ALIMOV's Publications

[1] Alimov A.R. Chebyshev sets in linear 2-spaces with non-symmetric unit ball // in ``Proceedings of 7th Saratov's Winter Workshop, 30 January --- 4~February 1994, dedicated to the memory of Prof. A.A. Privalov''. Mezhv. Sb. Nauchn. Trudov. Part.2. Saratov Univ. Press., 1995. p91--93. (in Russian).

[2] Alimov A.R. Quantity and connectedness of sun's complements // Prooceedings of Voronezh's Workshop on Functional Analysis and Approximation Theory, winter 1995; p. 11. (in Russian).

[3] Alimov A.R. A number of connected components of suns complement / Proceedings of the XIX Workshop on Function Theory (Beloretsk, 1994) // East J. Approx. 1 (1995), no. 4, 419--429.

[4] Alimov A.R. Chebyshev sets' complements / Proceedings of the XX Workshop on Function Theory (Moscow, 1995) // East J. Approx. 2(1996), no.2, 215--232.

[5] Alimov A.R. Letter to the editors: ``On a property of a circle in a two-dimensional Banach space" [Mat. Zametki 55 (1994), no.1, 79--83, p.157; MR 95c:52007] by I. V. Raskina. (Russian) Mat. Zametki 58 (1995), no.3, 477; translation in Math. Notes 58 (1995), no.3-4, p.~1010(1996) 52A21

[6] Alimov A.R. On the number of connected components of Chebyshev sets' complement // Proc. of Internat. conference on Approximation of functions dedicated to the memory of Prof. P.P.Korovkin. Kaluga, 26--29 June, 1996. Part.1. Kaluga, 1996 p.10--11. (in Russian).

[7] Alimov A.R. Chebyshev set's complement // in Abstracts of ``3rd Int. Conf. on Functional Analysis and Approximation Theory, Acquafredda di Maratea, Italy, Sept. 23--28, 1996''. p.~19.

[8] Alimov A.R.} Chebyshev compacts on a plane // Proc. Steklov Matem. Inst. 129, (1997) 8--19 (in Russian); Eng.\ Transl.: 8--26.

[9] Alimov A.R. Approximative properties of sets in linear spaces with non-symmetric sphere. Dissertation (Moscow State University, 1997).

[10] Alimov A.R., Karlov M.I. Sets with external Chebyshev layer (to appear).

[11] Alimov A.R. Solstice property for a system of 2-spaces // East. J.~Appr. bf 4 (1998), no.4, 25--34.

[12] Alimov A.R., Berens H. Examples of Chebyshev sets in matrix spaces // J.Appr. Theory (to appear in J.Appr.Theory).

[13] Alimov A.R. Are all Chebyshev sets convex? // Matematicshekoe Prosveschenie (Math. Education), ser.3, 3 (1998),155-172.

Summaries

A number of connected components of suns' complement. [3, 2].

We prove that a sun in an $n$-dimensional Banach space has at most $2^n$ connected components in its complement. We also give a characterisation (in terms of geometric properties of the unit sphere) of those normed linear spaces in which a sun exists with preassigned number of connected components in its complements. Similar problems are solved for linear spaces with a non-symmetric sphere.

We prove that a Chebyshev sets in an $n$-dimensional space has at most $2^n$ connected components in its complement. We also give necessary conditions and sufficient conditions for a space to contain a Chebyshev set with preassigned number of connected components in its complement.

Chebyshev sets in linear 2-spaces with non-symmetric unit ball. [1].

We give a characterisation of systems of 2-dimensional normed linear spaces in which every Chebyshev set with respect to the system of spaces is convex. Similar problem is solved for spaces with non-symmetric distance.

Chebyshev compacts on the plane. [8]

We characterise those 2-dimensional spaces $X$ and topological compacts $K$ for which $K$ may be homeomorphically embedded in~$X$ as a Chebyshev set (a~sun).

Solstice property for a system of 2-spaces. [11].

We prove that a sun over a system of 2-spaces $(X,\rho_i)_{i\in I}$ with non-symmetric norms~$\rho_i$ preserves its property to be a sun in the space $(X,\rho)$ with the (non-symmetric) norm $\rho=\sup_{i\in I} \rho_i$. Similar results are obtained for the Chebyshev sets.

Examples of Chebyshev sets in matrix spaces. [12] (with H.~Berens).

Let $A$ be a matrix in $\Cnn$ and let $U\Sigma V^*$ be its singular value decomposition, where $\Sigma=\roman{diag}(\sigma_1(S),\dots, \sigma_n(S)$. The authors prove that for each $1\le k\le n$ the set

$$

\{S\in\Cnn\,:\,

\sum_{1\le j_1<\dotsc<j_k\le n}

\sigma_{j_1}(S)\sigma_{j_2}(S)\cdots\sigma_{j_k}(S)\le 1\}

$$

is a Chebyshev set in $\Cnn$ endowed with the spectral norm and that the metric projection is globally Lipschitz-continuous.

Similar results are obtained for the spaces of Hermitian and symmetric matrices.

Sets with external Chebyshev layer. [11] (with M.I.~Karlov).

Let $X$ be a normed linear space. A set $M\subset X$ is said to be a set with {\it external Chebyshev layer}, if there is $a\ge 0$ such that $x\in X$, $\rho(x,M)>a$, implies $x$ has a unique element of best approximation from~$M$. We give a characterisation of sets with external Chebyshev layer in finite-dimensional spaces.

Are all Chebyshev sets convex? [13]

In 1935 L.N.H.~Bunt proved that in $\bR^n$ every Chebyshev set (i.e., a set, such that each point has a unique element of best approximation from this set) is closed and convex (clearly, the inverse statement is also true). In this paper we give 5~different proofs of Bunt's theorem. Similar results are discussed for arbitrary Banach spaces.