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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Stable and uniformly stable unit balls in Banach spaces


Author: Antonio Suárez Granero
Journal: Trans. Amer. Math. Soc. 330 (1992), 677-695
MSC: Primary 46B20; Secondary 46E40
DOI: https://doi.org/10.1090/S0002-9947-1992-1031977-1
MathSciNet review: 1031977
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Abstract: Let $ X$ be a Banach space with closed unit ball $ {B_X}$ and, for $ x \in X$, $ r \geq 0$, put $ B(x;r)= \{ u \in X:\vert\vert u - x\vert\vert \leq r\} $ and $ V(x,r)= {B_X} \cap B(x;r)$. We say that $ {B_X}$ (or in general a convex set) is stable if the midpoint map $ {\Phi _{1/2}}:{B_X} \times {B_X} \to {B_X}$, with $ {\Phi _{1/2}}(u,\upsilon)= \frac{1}{2}(u + \upsilon)$, is open. We say that $ {B_X}$ is uniformly stable (US) if there is a map $ \alpha :(0,2] \to (0,2]$, called a modulus of uniform stability, such that, for each $ x,y \in {B_X}$ and $ r \in (0,2],V(\frac{1} {2}(x + y);\alpha (r)) \subseteq \frac{1} {2}(V(x;r) + V(y;r))$. Among other things, we see: (i) if $ \dim X \geq 3$, then $ X$ admits an equivalent norm such that $ {B_X}$ is not stable; (ii) if $ \dim X < \infty $, $ {B_X}$ is stable iff $ {B_x}$ is US; (iii) if $ X$ is rotund, $ X$ is uniformly rotund iff $ {B_X}$ is US; (iv) if $ X$ is $ 3.2.{\text{I.P}}$, $ {B_X}$ is US and $ \alpha (r)= r/2$ is a modulus of US; (v) $ {B_X}$ is US iff $ {B_{{X^{ \ast \ast }}}}$ is US and $ X$, $ {X^{ \ast \ast}}$ have (almost) the same modulus of US; (vi) $ {B_X}$ is stable (resp. US) iff $ {B_{C(K,X)}}$ is stable (resp. US) for each compact $ K$ iff $ {B_{A(K,X)}}$ is stable (resp. US) for each Choquet simplex $ K$; (vii) $ {B_X}$ is stable iff $ {B_{{L_p}(\mu,X)}}$ is stable for each measure $ \mu $ and $ 1 \leq p < \infty $.


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DOI: https://doi.org/10.1090/S0002-9947-1992-1031977-1
Keywords: Stable and uniformly stable sets, unit ball, Banach spaces
Article copyright: © Copyright 1992 American Mathematical Society