Kania, Tomasz and Kochanek, Tomasz (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 ISSN 0022-247X.

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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.

Item Type:	Journal Article
Subjects:	Q Science > QA Mathematics
Divisions:	Faculty of Science, Engineering and Medicine > Science > Mathematics
Library of Congress Subject Headings (LCSH):	Banach spaces, Vector spaces
Journal or Publication Title:	Journal of Mathematical Analysis and Applications
Publisher:	Academic Press
ISSN:	0022-247X
Official Date:	February 2017
Dates:	Date Event February 2017 Published 17 September 2016 Available 16 September 2016 Accepted 30 March 2016 Submitted
Volume:	446
Number:	2
Page Range:	pp. 1219-1229
DOI:	10.1016/j.jmaa.2016.09.030
Status:	Peer Reviewed
Publication Status:	Published
Access rights to Published version:	Restricted or Subscription Access
Date of first compliant deposit:	24 November 2016
Date of first compliant Open Access:	17 September 2017
Funder:	European Research Council (ERC), Poland. Ministerstwo Nauki i Szkolnictwa Wyższego [Ministry of Science and Higher Education]
Grant number:	Grant Agreement No. 291497 (ERC), Project No. IP2012011072.
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