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Extension of Simons' inequality
Authors:Kersti Kivisoo   Eve Oja
Affiliation:Faculty of Mathematics and Computer Science, Tartu University, J. Liivi 2, EE-50409 Tartu, Estonia ; Faculty of Mathematics and Computer Science, Tartu University, J. Liivi 2, EE-50409 Tartu, Estonia
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 $f: Ctimes S longrightarrow mathbb{R}$ be a bounded function such that the functions $f(,cdot ,,,t):Clongrightarrow mathbb{R}$ 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 $sin S.$ Let $(x_n)$ be a sequence in $C$. Assume that, for every $x in C_1 =left{sum_{n=1}^{infty}lambda_n,x_n,:quad lambda_ngeq 0,,sum_{n=1}^{infty}lambda_n=1,right}$, there exists $t in T$ satisfying $f(x,,t)=sup_{sin S} f(x,,s)$. Then

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

If $-C_1subset C$, then the set $C_1$ in the above inequality can be replaced by ${rm conv}{x_1, x_2, ldots}$.

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