This is an announcement for the paper "Steinhaus' lattice-point problem for Banach spaces" by Tomasz Kania and Tomasz Kochanek.

Abstract: Given a positive integer $n$, one may find a circle surrounding exactly $n$ points of the integer lattice. This classical geometric fact due to Steinhaus has been recently extended to Hilbert spaces by Zwole'{n}ski, who replaced the integer lattice by any infinite set which intersects every ball in at most finitely many points. We investigate the norms satisfying this property, which we call (S), and show that all strictly convex norms have (S). Nonetheless, we construct a norm in dimension three which has (S) but fails to be strictly convex. Furthermore, the problem of finding an equivalent norm enjoying (S) is studied. With the aid of measurable cardinals, we prove that there exists a Banach space having (S) but with no strictly convex renorming.

Archive classification: math.FA

Submitted from: t.kania@lancaster.ac.uk

The paper may be downloaded from the archive by web browser from URL

http://front.math.ucdavis.edu/1302.6443

or

http://arXiv.org/abs/1302.6443