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(·) belongs to BLO1/ log then for the norm of corresponding variable exponent Lebesgue space we have the following lower estimate ∥ ∥∑χQ‖ f χQ‖p(·)/‖χQ‖p(·) ∥ ∥ p(·) C‖ f‖p(·) 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(·) C ∥ ∥∑χQ‖ f χQ‖p(·)/‖χQ‖p(·) ∥ ∥ p(·) . Mathematics subject classification (2010): 42B35, 42B20, 46B45, 42B25.