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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: We prove the following extended version of Simons' inequality and present its applications. Let $S$ be a set and $T$ be a subset of $S$. Let $C$ be a subset of a Hausdorff topological vector space which is invariant under infinite convex combinations. Let $f: C\times S \longrightarrow \mathbb{R}$ be a bounded function such that the functions $f(\,\cdot \,,\,t):C\longrightarrow \mathbb{R}$ are convex for all $t \in T$ and $f(\lambda x,\,s)=\lambda f(x,\,s)$ whenever $\lambda >0$, $x,\,\lambda x \in C$ and $s\in S.$ Let $(x_n)$ be a sequence in $C$. Assume that, for every $x \in C_1 =\left\{\sum_{n=1}^{\infty}\lambda_n\,x_n\,:\quad \lambda_n\geq 0,\,\sum_{n=1}^{\infty}\lambda_n=1\,\right\}$, there exists $t \in T$ satisfying $f(x,\,t)=\sup_{s\in S} f(x,\,s)$. Then

\begin{displaymath}\inf_{x\in C_1}\sup_{s\in S}f(x,\,s) \leq \sup_{t\in T}\limsup_{n}f(x_n,\,t).\end{displaymath}

If $-C_1\subset C$, then the set $C_1$ in the above inequality can be replaced by ${\rm conv}\{x_1, x_2, \ldots\}$.


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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.

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