Geodesic
Existence of untrivial compact Tchebycheff sets in the spaces~$L_\varphi$
Sbornik. Mathematics, Tome 69 (1991) no. 2, pp. 431-444

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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. 
@article{SM_1991_69_2_a6,
     author = {D. G. Kamuntavichius},
     title = {Existence of untrivial compact {Tchebycheff} sets in the spaces~$L_\varphi$},
     journal = {Sbornik. Mathematics},
     pages = {431--444},
     publisher = {mathdoc},
     volume = {69},
     number = {2},
     year = {1991},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_1991_69_2_a6/}
}
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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/