Extension of Simons' inequality

Authors:
Kersti Kivisoo and Eve Oja

Journal:
Proc. Amer. Math. Soc. **133** (2005), 3485-3496

MSC (2000):
Primary 39B62, 46A55, 46B20, 54C30

DOI:
https://doi.org/10.1090/S0002-9939-05-08267-5

Published electronically:
June 28, 2005

MathSciNet review:
2163583

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Abstract | References | Similar Articles | Additional Information

Abstract: We prove the following extended version of Simons' inequality and present its applications. Let be a set and be a subset of . Let be a subset of a Hausdorff topological vector space which is invariant under infinite convex combinations. Let be a bounded function such that the functions are convex for all and whenever , and Let be a sequence in . Assume that, for every , there exists satisfying . Then

If , then the set in the above inequality can be replaced by .

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Additional Information

**Kersti Kivisoo**

Affiliation:
Faculty of Mathematics and Computer Science, Tartu University, J. Liivi 2, EE-50409 Tartu, Estonia

Email:
kersti.kivisoo@mail.ee

**Eve Oja**

Affiliation:
Faculty of Mathematics and Computer Science, Tartu University, J. Liivi 2, EE-50409 Tartu, Estonia

Email:
eveoja@math.ut.ee

DOI:
https://doi.org/10.1090/S0002-9939-05-08267-5

Keywords:
Simons' inequality,
convex sets in topological vector spaces,
convex functions,
uniformly convergent convex combinations,
Banach space geometry.

Received by editor(s):
July 2, 2004

Published electronically:
June 28, 2005

Additional Notes:
This research was partially supported by Estonian Science Foundation Grant 5704

Communicated by:
Jonathan M. Borwein

Article copyright:
© Copyright 2005
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.