Geodesic
$2$-Chebyshev Subspaces in the Spaces~$L_1$ and~$C$
Matematičeskie zametki, Tome 91 (2012) no. 6, pp. 819-831.

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The $2$-uniqueness subspaces and the finite-dimensional $2$-Chebyshev subspaces of the space $C$ of functions continuous on a Hausdorff compact set and of the space $L_1$ of functions Lebesgue integrable on a set of $\sigma$-finite measure are described. These descriptions are analogs of the well-known Haar and Phelps theorems for ordinary Chebyshev subspaces.
Keywords: Banach space, Hilbert space, $2$-Chebyshev subspace, $2$-uniqueness subspace, $2$-existence subspace, the space $L_1$ of Lebesgue integrable functions. 
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P. A. Borodin. $2$-Chebyshev Subspaces in the Spaces~$L_1$ and~$C$. Matematičeskie zametki, Tome 91 (2012) no. 6, pp. 819-831. http://geodesic.mathdoc.fr/item/MZM_2012_91_6_a2/

[1] N. V. Efimov, S. B. Stechkin, “Nekotorye svoistva chebyshevskikh mnozhestv”, DAN SSSR, 118:1 (1958), 17–19 | MR | Zbl

[2] V. S. Balaganskii, L. P. Vlasov, “Problema vypuklosti chebyshevskikh mnozhestv”, UMN, 51:6(312) (1996), 125–188 | MR | Zbl

[3] P. A. Borodin, “Vypuklost 2-chebyshevskikh mnozhestv v gilbertovom prostranstve”, Vestn. Mosk. un-ta. Ser. 1. Matem., mekh., 2008, no. 3, 16–19 | MR | Zbl

[4] P. A. Borodin, “O vypuklosti $N$-chebyshevskikh mnozhestv”, Teoriya priblizhenii, Tezisy dokladov mezhdunarodnoi konferentsii, posvyaschennoi 90-letiyu S. B. Stechkina, M., 2010, 11–12

[5] A. L. Garkavi, “Teoriya nailuchshego priblizheniya v lineinykh normirovannykh prostranstvakh”, Itogi nauki. Ser. Matematika. Mat. anal. 1967, VINITI, M., 1969, 75–132 | MR | Zbl

[6] I. Singer, Best Approximation in Normed Linear Spaces by Elements of Linear Subspaces, Grundlehren Math. Wiss., 171, Springer-Verlag, Berlin, 1970 | MR | Zbl

[7] G. Sh. Rubinshtein, “Ob odnoi ekstremalnoi zadache v lineinom normirovannom prostranstve”, Sib. matem. zhurn., 6:3 (1965), 711–714 | MR | Zbl

[8] N. Danford, Dzh. Shvarts, Lineinye operatory. T. 1. Obschaya teoriya, IL, M., 1962 | MR | Zbl

[9] E. W. Cheney, D. E. Wulbert, “The existence and unicity of best approximations”, Math. Scand., 24:1 (1969), 113–140 | MR | Zbl

[10] R. R. Phelps, “Čebyšev subspaces of finite codimension in $C(X)$”, Pacific J. Math., 13:2 (1963), 647–655 | MR | Zbl

[11] A. Haar, “Die Minkowskische Geometrie und die Annäherung an stetige Funktionen”, Math. Ann., 78:1-4 (1917), 294–311 | DOI | MR | Zbl

[12] P. S. Aleksandrov, Vvedenie v teoriyu mnozhestv i obschuyu topologiyu, Fizmatlit, M., 2009 | MR

[13] J. C. Mairhuber, “On Haar's theorem concerning Chebychev approximation problems having unique solutions”, Proc. Amer. Math. Soc., 7:4 (1956), 609–615 | MR | Zbl

[14] R. R. Phelps, “Čebyšev subspaces of finite dimension in $L_1$”, Proc. Amer. Math. Soc., 17:3 (1966), 646–652 | MR | Zbl

[15] R. R. Phelps, “Uniqueness of Hahn-Banach extensions and unique best approximation”, Trans. Amer. Math. Soc., 95:2 (1960), 238–255 | MR | Zbl

[16] M. Krein, “$L$-problema v abstraktnom lineinom normirovannom prostranstve”: N. Akhiezer, M. Krein, O nekotorykh voprosakh teorii momentov, GONTI, Kharkov, 1938

[17] P. D. Morris, “Chebyshev subspaces of $L_1$ with linear metric projection”, J. Approx. Theory, 29:3 (1980), 231–234 | DOI | MR | Zbl

[18] P. A. Borodin, “O lineinosti operatora metricheskogo proektirovaniya na chebyshevskie podprostranstva v prostranstvakh $L_1$ i $C$”, Matem. zametki, 63:6 (1998), 812–820 | MR | Zbl

[19] V. I. Bogachev, Osnovy teorii mery, T. 1, NITs RKhD, Izhevsk, 2003