Banach October 2016

banach@mathdept.okstate.edu

2 participants
22 discussions

Abstract of a paper by Grigiry Ivanov, Horst Martini
by Bentuo Zheng (bzheng) 01 Oct '16

01 Oct '16

This is an announcement for the paper “New Moduli for Banach Spaces” by Grigiry Ivanov<https://arxiv.org/find/math/1/au:+Ivanov_G/0/1/0/all/0/1>, Horst Martini<https://arxiv.org/find/math/1/au:+Martini_H/0/1/0/all/0/1>. Abstract: Modifying the moduli of supporting convexity and supporting smoothness, we introduce new moduli for Banach spaces which occur, e.g., as lengths of catheti of right-angled triangles (defined via so-called quasi-orthogonality). These triangles have two boundary points of the unit ball of a Banach space as endpoints of their hypotenuse, and their third vertex lies in a supporting hyperplane of one of the two other vertices. Among other things it is our goal to quantify via such triangles the local deviation of the unit sphere from its supporting hyperplanes. We prove respective Day-Nordlander type results, involving generalizations of the modulus of convexity and the modulus of Bana\'{s}. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1609.01587

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Announcement of a paper by Luigi Ambrosio, Daniele Puglisi
by Bentuo Zheng (bzheng) 01 Oct '16

01 Oct '16

This is an announcement for the paper “Linear extension operators between spaces of Lipschitz maps and optimal transport” by Luigi Ambrosio<https://arxiv.org/find/math/1/au:+Ambrosio_L/0/1/0/all/0/1>, Daniele Puglisi<https://arxiv.org/find/math/1/au:+Puglisi_D/0/1/0/all/0/1>. Abstract: Motivated by the notion of $K$-gentle partition of unity introduced in [12] and the notion of $K$-Lipschitz retract studied in [17], we study a weaker notion related to the Kantorovich-Rubinstein transport distance, that we call $K$-random projection. We show that $K$-random projections can still be used to provide linear extension operators for Lipschitz maps. We also prove that the existence of these random projections is necessary and sufficient for the existence of weak$^*$ continuous operators. Finally we use this notion to characterize the metric spaces $(X, d)$ such that the free space $F(X)$ has the bounded approximation propriety. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1609.01450

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Banach October 2016