Every nonreflexive Banach lattice has the packing constant equal to 1/2

Kottman [9] has proved that any $P$-convex Banach space $X$ is reflexive. In the case when $X$ is a Banach lattice our result says more. It says that any Banach lattice $X$ with $\Lambda(X) < 1/2$ is reflexive. This result generalizes the results of Berezhnoi [2] who proved that $\Lambda(\Lambda(\varphi)) = \Lambda(M(\varphi)) = 1/2$ for nonreflexive Lorentz space $\Lambda(\varphi)$ and Marcinkiewicz space $M(\varphi)$. It is proved also that for any Banach lattice $X$ such that its subspace $X_a$ of order continuous elements is nontrivial we have $\Lambda(X) = \Lambda(X_a)$. It is noted also that Orlicz sequence space $l^\Phi$ is reflexive iff $\Lambda(l^\Phi) < 1/2$.