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THE REFLEXIVITY OF 2-KNOTS IN S2 × S2
Preview AbstractWe consider a locally flat 2-sphere embedded in S2 × S2, which is referred to as a 2-knot in S2 × S2, and we study the problem whether a 2-knot in S2 × S2 is determined by its exterior. In this paper, we show that for almost homology classes ξ, 2-knots in S2 × S2 representing ξ are determined by their exteriors.
On metric characterizations of the Radon–Nikodým and related properties of Banach spaces
Preview AbstractWe find a class of metric structures which do not admit bilipschitz embeddings into Banach spaces with the Radon–Nikodým property. Our proof relies on Chatterji's (1968) martingale characterization of the RNP and does not use the Cheeger's (1999) metric differentiation theory. The class includes the infinite diamond and both Laakso (2000) spaces. We also show that for each of these structures there is a non-RNP Banach space which does not admit its bilipschitz embedding.
We prove that a dual Banach space does not have the RNP if and only if it admits a bilipschitz embedding of the infinite diamond.
The paper also contains related characterizations of reflexivity and the infinite tree property.