Existence of untrivial compact Tchebycheff sets in the spaces $L_\varphi$

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.