  In the present paper we investigate some geometrical properties of the norms
in Banach function spaces. Particularly there is shown that if exponent
$1/p(\cdot)$ belongs to $BLO^{1/\log}$ then for the norm of corresponding
variable exponent Lebesgue space we have the following lower estimate
$$\left\|\sum
\chi_{Q}\|f\chi_{Q}\|_{p(\cdot)}/\|\chi_{Q}\|_{p(\cdot)}\right\|_{p(\cdot)}\leq
C\|f\|_{p(\cdot)}$$ where $\{Q\}$ defines disjoint partition of $[0;1]$. Also
we have constructed variable exponent Lebesgue space with above property which
does not possess following upper estimation $$\|f\|_{p(\cdot)}\leq C\left\|\sum
\chi_{Q}\|f\chi_{Q}\|_{p(\cdot)}/\|\chi_{Q}\|_{p(\cdot)}\right\|_{p(\cdot)}. $$
