[Settfa] Tuesday 14/11, Tommaso Russo

[kubis at math.cas.cz]

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 Tuesday 14th November, 10:00am 
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 Speaker:Tommaso Russo, Dipartimento di matematica, Università degli Studi di Milano
 Title: Symmetrically separated sequences in the unit sphere of a Banach space

 Abstract  
There are several results in the literature concerning the existence of an infinite subset $A$ of the unit sphere of a Banach space such that $\| x-y\| \geq 1+\varepsilon$ for distinct $x,y$ in $A$. One of the famous results in this direction is the Elton--Odell theorem; further results, where quantitative estimates on $\varepsilon$ are given, are also present in the literature.
In this talk, based on a joint work with Petr Hájek and Tomasz Kania, we investigate the symmetric analogue of the above results, namely we study the circumstances under which a set $A$ may be found such that $\|x\pm y\| \geq 1+\varepsilon$ for distinct elements $x,y\in A$. In particular, we prove the symmetric version of Kottman's theorem, that is to say, we demonstrate that the unit sphere of an infinite-dimensional Banach space contains an infinite subset $A$ with the property that $\|x\pm y\| > 1$ for distinct elements $x,y\in A$, thereby answering a question of Castillo. In the case where $X$ contains an unconditional basic sequence, the set $A$ may be chosen in a way that $\|x\pm y\| \geq 1+\varepsilon$ for some $\varepsilon > 0$ and distinct $x,y\in A$. Under additional structural properties of $X$, such as non-trivial cotype, we obtain quantitative estimates for the said $\varepsilon$. Certain renorming results are also presented.

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