Extension of Simons' inequality

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

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