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 be a Banach space with closed unit ball and, for , , put and . We say that (or in general a convex set) is *stable* if the midpoint map , with , is open. We say that is *uniformly stable* (US) if there is a map , called a *modulus of uniform stability*, such that, for each and . Among other things, we see: (i) if , then admits an equivalent norm such that is not stable; (ii) if , is stable iff is US; (iii) if is rotund, is uniformly rotund iff is US; (iv) if is , is US and is a modulus of US; (v) is US iff is US and , have (almost) the same modulus of US; (vi) is stable (resp. US) iff is stable (resp. US) for each compact iff is stable (resp. US) for each Choquet simplex ; (vii) is stable iff is stable for each measure and .

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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