Extension of Simons’ inequality

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 \[ \inf _{x\in C_1}\sup _{s\in S}f(x, s) \leq \sup _{t\in T}\limsup _{n}f(x_n, t).\] If $-C_1\subset C$, then the set $C_1$ in the above inequality can be replaced by $\textrm {conv}\{x_1, x_2, \ldots \}$.