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 This improves an earlier unpublished result of E. Matoušková.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 ℓ₁ and L₁ can be decomposed as unions of a ball small set and an Aronszajn null set.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-cm108-1-13,
author = {Jakub Duda},
title = {On a decomposition of Banach spaces},
journal = {Colloquium Mathematicae},
volume = {107},
year = {2007},
pages = {147-157},
zbl = {1118.28001},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm108-1-13}
}
Jakub Duda. On a decomposition of Banach spaces. Colloquium Mathematicae, Tome 107 (2007) pp. 147-157. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm108-1-13/