Geodesic
Existence of untrivial compact Tchebycheff sets in the spaces $L_\varphi$
Sbornik. Mathematics, Tome 69 (1991) no. 2, pp. 431-444 Cet article a éte moissonné depuis la source Math-Net.Ru

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It is proved that if $(T,\Omega,\mu)$ is a nonatomic measure space and $\varphi$ an even function nondecreasing on $[0,\infty)$ and such that $\varphi(0)=0$, $\varphi(u)>0$ for $u>0$, and $\varphi(u_1+u_2)<\varphi(u_1)+\varphi(u_2)$ for all $u_1,u_2>0$, then the space $L_\varphi(T,\Omega,\mu)$ does not contain boundedly compact Tchebycheff sets with more than one point. 
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     author = {D. G. Kamuntavichius},
     title = {Existence of untrivial compact {Tchebycheff} sets in the spaces~$L_\varphi$},
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D. G. Kamuntavichius. Existence of untrivial compact Tchebycheff sets in the spaces $L_\varphi$. Sbornik. Mathematics, Tome 69 (1991) no. 2, pp. 431-444. http://geodesic.mathdoc.fr/item/SM_1991_69_2_a6/

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

[2] Vlasov L. P., “O chebyshevskikh mnozhestvakh”, DAN SSSR, 173:3 (1967), 443–449 | MR

[3] Efimov N. V., Stechkin S. B., “Chebyshevskie mnozhestva v banakhovykh prostranstvakh”, DAN SSSR, 121:4 (1958), 582–585 | MR | Zbl

[4] Efimov N. V., Stechkin S. B., “Approksimativnaya kompaktnost i chebyshevskie mnozhestva”, DAN SSSR, 140:3 (1961), 522–524 | MR | Zbl

[5] Klee V. L., “A characterization of convex sets”, Amer. Math. Monthly, 56 (1949), 247–249 | DOI | MR | Zbl

[6] Klee V. L., “Convex bodies and periodic homeomorphisms in Hilbert space”, Trans. Amer. Math. Soc., 74 (1953), 10–43 | DOI | MR | Zbl

[7] Vlasov L. P., “Approksimativnye svoistva mnozhestv v lineinykh normirovannykh prostranstvakh”, UMN, 28:6 (1973), 3–66 | MR | Zbl

[8] Garkavi A. L., “Teoriya nailuchshego priblizheniya v lineinykh normirovannykh prostranstvakh”, Itogi nauki. Matem. analiz. 1967, VINITI, M., 1969, 75–132

[9] Klee V. L., “Convexity of Chebyshev Sets”, Math. Annalen, 142:3 (1961), 292–304 | DOI | MR | Zbl

[10] Tsarkov I. G., “O kompaktnykh i slabo kompaktnykh chebyshevskikh mnozhestvakh v lineinykh normirovannykh prostranstvakh”, DAN SSSR, 300:1 (1988), 41–42 | MR

[11] Kamuntavichyus D. G., “Suschestvovanie chebyshevskikh kompaktov v separabelnom banakhovom prostranstve”, XXIX konferentsiya Litov. matem. obsch., Tez. dokl., Vilnyus, 1988, 143–144

[12] Danford N., Shvarts Dzh., Lineinye operatory. Obschaya teoriya, IL, M., 1962

[13] Valentine F. A., Convex Sets, Mo Graw-Hill Book Company, New York, 1964 | MR

[14] Kamuntavichyus D. G., “Struktura metricheskoi proektsii na odnomernoe podprostranstvo v $L_{\varphi}$”, Litov. matem. sb., 29:3 (1989), 464–473 | MR | Zbl