A Universal Inequality for Stability of Coarse Lipschitz Embeddings
A Universal Inequality for Stability of Coarse Lipschitz Embeddings
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DOI：
10.1007/s10114-023-2136-4
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发表时间：
2023-09
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影响因子：
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通讯作者：
Duan Xu Dai;Ji Chao Zhang;Quan Qing Fang;Long Fa Sun;Ben Tuo Zheng
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作者：
Duan Xu Dai;Ji Chao Zhang;Quan Qing Fang;Long Fa Sun;Ben Tuo Zheng
设X和Y是两个点度量空间。本文将Cheng-Dong-Zhang定理推广到粗Lipschitz嵌入：若f：X→ Y是标准粗Lipschitz嵌入，则对每个x * ∈ Lip 0（X），存在α，γ> 0仅依赖于f和Qx *∈ Lip 0（Y），使得|Qx*f（x）−x*（x）|≤γ <$x*<$Lip，其中x ∈X.研究了度量空间对的粗稳定性.这可以看作是钱问题的一个粗略的版本。作为应用，我们给候选人的否定答案的一个58岁的问题，Lindenstrauss问是否每个Banach空间是一个Lipschitz收缩的双对偶。事实上，我们表明，X不是Lipschitz收缩的双对偶，如果X是一个普遍左粗稳定的空间，但不是一个绝对的基数Lipschitz收缩。如果存在一个具有RNP的泛右粗稳定的Banach空间，但不同构于任何Hilbert空间，则对于可分空间，该问题也有一个否定解。
LetXandYbe two pointed metric spaces. In this article, we give a generalization of the Cheng–Dong–Zhang theorem for coarse Lipschitz embeddings as follows: Iff:X→Yis a standard coarse Lipschitz embedding, then for eachx* ∈ Lip0(X) there existα, γ> 0 depending only onfandQx*∈ Lip0(Y) withsuch that |Qx*f(x)−x*(x)|≤γ‖x*‖Lip,forallx∈X.Coarse stability for a pair of metric spaces is studied. This can be considered as a coarse version of Qian Problem. As an application, we give candidate negative answers to a 58-year old problem by Lindenstrauss asking whether every Banach space is a Lipschitz retract of its bidual. Indeed, we show thatXis not a Lipschitz retract of its bidual ifXis a universally left-coarsely stable space but not an absolute cardinality-Lipschitz retract. If there exists a universally right-coarsely stable Banach space with the RNP but not isomorphic to any Hilbert space, then the problem also has a negative answer for a separable space.