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The Bishop--Phelps--Bollob'as property for Lipschitz maps | | Rafael Chiclana
; Miguel Martin
; | | Date: |
9 Jan 2019 | | Abstract: | In this paper, we introduce and study a Lipschitz version of the
Bishop-Phelps-Bollob’as property (Lip-BPB property). This property deals with
the possibility to make a uniformly simultaneous approximation of a Lipschitz
map $F$ and a pair of points at which $F$ almost attains its norm by a
Lipschitz map $G$ and a pair of points such that $G$ strongly attains its norm
at the new pair of points. We first show that if $M$ is a finite pointed metric
space and $Y$ is a finite-dimensional Banach space, then the pair $(M,Y)$ has
the Lip-BPB property, and that both finiteness are needed. Next, we show that
if $M$ is a uniformly Gromov concave pointed metric space (i.e. the molecules
of $M$ form a set of uniformly strongly exposed points), then $(M,Y)$ has the
Lip-BPB property for every Banach space $Y$. We further prove that this is the
case of finite concave metric spaces, ultrametric spaces, and H"older metric
spaces. The extension of the Lip-BPB property from $(M,mathbb{R})$ to some
Banach spaces $Y$, the relationship with absolute sums, and some results only
valid for compact Lipschitz maps, are also discussed. | | Source: | arXiv, 1901.2956 | | Services: | Forum | Review | PDF | Favorites | |
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