Steinhaus' lattice-point problem for Banach spaces

Loading...
Thumbnail Image
Files
4071.pdf(386.08 KB)
Accepted Version
Date
2017-02-15
Authors
Kania, Tomasz
Kochanek, Tomasz
Journal Title
Journal ISSN
Volume Title
Publisher
Elsevier Ltd.
Research Projects
Organizational Units
Journal Issue
Abstract
Steinhaus proved that given a positive integer n, one may find a circle surrounding exactly n points of the integer lattice. This statement has been recently extended to Hilbert spaces by Zwoleński, who replaced the integer lattice by any infinite set that intersects every ball in at most finitely many points. We investigate Banach spaces satisfying this property, which we call (S), and characterise them by means of a new geometric property of the unit sphere which allows us to show, e.g., that all strictly convex norms have (S), nonetheless, there are plenty of non-strictly convex norms satisfying (S). We also study the corresponding renorming problem.
Description
Keywords
Steinhaus' problem , Lattice points , Strictly convex space
Citation
Kania, T. and Kochanek, T. (2017) ‘Steinhaus' lattice-point problem for Banach spaces’, Journal of Mathematical Analysis and Applications, 446(2), pp. 1219-1229. doi:10.1016/j.jmaa.2016.09.030