Steinhaus' lattice-point problem for Banach spaces

TL;DR

This paper explores a geometric property in Banach spaces related to lattice-point problems, characterizing spaces with this property and examining the impact of convexity and renorming.

Contribution

It introduces a new geometric property (S) in Banach spaces, characterizes spaces satisfying it, and analyzes the role of convexity and renorming in this context.

Findings

All strictly convex norms satisfy property (S).

Many non-strictly convex norms also satisfy (S).

Characterization of Banach spaces with property (S) via geometric properties of the unit sphere.

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 Zwole\'{n}ski, 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.