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Steinhaus' lattice-point problem for Banach spaces
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.
Steinhaus' problem , Lattice points , Strictly convex space
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