[Settfa] Friday 21/10, Tomasz Kania

[kubis at math.cas.cz]

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 Friday 21st October, 11:00am 
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 Speaker:Tomasz Kania, University of Warwick, UK
 Title: Steinhaus lattice-point problem for Banach spaces

 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 where the integer lattice was replaced 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 we 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. Assuming that the Lebesgue measure can be extended to a measure defined on the power set of the reals, we construct a Banach space that has (S) but it does not have a strictly convex renorming. 

 For more information see the seminar web page at 
 https://calendar.math.cas.cz/set-theory-and-analysis-actual .