In spite of the Lebesgue density theorem, there is a positive $${\delta}$$ δ such that, for every measurable set $${A \subset \mathbb{R}}$$ A ⊂ R with $${\lambda (A) > 0}$$ λ ( A ) > 0 and $${\lambda (\mathbb{R} \setminus A) > 0}$$ λ ( R \ A ) > 0 , there is a point at which both the lower densities of $${A}$$ A and of the complement of $${A}$$ A are at least $${\delta}$$ δ . The problem of determining the supremum of possible values of this $${\delta}$$ δ was studied by V. I. Kolyada, A. Szenes and others. It seems that the authors considered this quantity a feature of density. We show that it is connected rather with a choice of a differentiation basis.