On the Exact Value of Packing Spheres in a Class of Orlicz Function Spaces

Main result: the packing constants of Orlicz function spaces $L^{(\Phi)}[0,1]$ and $L^{\Phi}[0,1]$ with Luxemburg and Orlicz norm have the exact value. (i) If $F_\Phi(t)=t\varphi(t)/\Phi(t)$ is decreasing, $1<C_\Phi< 2,$ then $$P(L^{(\Phi)}[0,1])=P(L^{\Phi}[0,1])=\frac{2^{1/C_\Phi}}{2+2^{1/C_\Phi}};$$ (ii) If $F_\Phi(t)$ is increasing, $C_\Phi> 2,$ then $$P(L^{(\Phi)}[0,1])=P(L^{\Phi}[0,1])=\frac{1}{1+2^{1/C_\Phi}},$$ where $C_\Phi=\lim\limits_{t\rightarrow\infty} F_\Phi(t)$.