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Abstract

By using D. Preiss' approach to a construction from a paper by J. Matoušek and E. Matoušková, and some results of E. Matoušková, we prove that we can decompose a separable Banach space with modulus of convexity of power type $p$ as a union of a ball small set (in a rather strong symmetric sense) and a set which is Aronszajn null. This improves an earlier unpublished result of E. Matoušková. As a corollary, in each separable Banach space with modulus of convexity of power type $p$, there exists a closed nonempty set $A$ and a Borel non-Haar null set $Q$ such that no point from $Q$ has a nearest point in $A$. Another corollary is that $\ell_1$ and $L_1$ can be decomposed as unions of a ball small set and an Aronszajn null set.