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Register a new userAn asymptotic property of Schachermayers space under renorming
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A Banach space X with closed unit ball B is said to have property 2-beta, repsectively 2-NUC if for every ep > 0, there exists delta > 0 such that for every ep-separated sequence (x_n) in the unit ball B, and every x in B, there are distinct indices m and n such that ||x + x_m + x_n|| < 3(1 - delta), respectively, ||x_m + x_n|| < 2(1 - delta). It is shown that a Banach space constructed by Schachermayer has property 2-beta but cannot be renormed to have property 2-NUC.
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We study the problem of improving the greedy constant or the democracy constant of a basis of a Banach space by renorming. We prove that every Banach space with a greedy basis can be renormed, for a given $vare>0$, so that the basis becomes $(1+vare)$-democratic, and hence $(2+vare)$-greedy, with respect to the new norm. If in addition the basis is bidemocratic, then there is a renorming so that in the new norm the basis is $(1+vare)$-greedy. We also prove that in the latter result the additional assumption of the basis being bidemocratic can be removed for a large class of bases. Applications include the Haar systems in $L_p[0,1]$, $1<p<infty$, and in dyadic Hardy space $H_1$, as well as the unit vector basis of Tsirelson space.
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