PACKING CONSTANTS IN ORLICZ-LORENTZ SEQUENCE SPACES

Abstract

We discussed the upper and lower bounds of packing constants in Orlicz-Lorentz sequence spaces equipped with both the Luxemburg norm and the Orlicz norm. Provided $\Phi \in \Delta_2(0)$, we showed that the Kottman constant of $\lambda_{\Phi,\omega}$ and $\lambda^o_{\Phi,\omega}$ satisfies $$\max \left\{ \frac{1}{\alpha_\Phi(0)}, \frac{1}{\alpha'_{\Phi,\omega}} \right\} \leq K\left(\lambda_{\Phi,\omega}\right) \leq \frac{1}{\tilde{\alpha}_{\Phi,\omega}},$$ $$ \max \left\{ \frac{1}{\alpha_\Phi(0)}, \frac{1}{\alpha''_{\Phi,\omega}} \right\} \leq K\left(\lambda^o_{\Phi,\omega}\right) \leq \frac{1}{\alpha^{\ast}_{\Phi}}. $$ As a corollary, the packing constant of Lorentz space $\lambda_{p,\omega}$ is $1/(1+2^{1-\frac{1}{p}})$.