An Abstract Variational Theorem

Let $(X, \|\cdot\|)$ be a Banach space and $f\colon X \to \mathbb{R} \cup \{\infty\}$ be a proper function. Then the Fenchel conjugate of $f$ is the function $f^*\colon X^* \to \mathbb{R} \cup \{\infty\}$ defined by, $$f^*(x^*):= \sup\{(x^*-f)(x):x \in X\}.$$ In this article we will prove a theorem more general than the following. Theorem: Let $f\colon X \to \mathbb{R} \cup \{\infty\}$ be a proper function on a Banach space $(X,\|\cdot\|)$. If there is a nonempty open subset $A$ of $\mathrm{Dom}(f^*)$ such that $\mathrm{argmax}(x^*-f) \not= \varnothing$ for each $x^* \in A$, then there is a dense and $G_\delta$ subset $R$ of $A$ such that $(x^*-f) \colon X \to \mathbb{R} \cup \{-\infty\}$ has a strong maximum for each $x^* \in R$. In addition, if $0 \in A$ and $0<\varepsilon$ then there is an $x^* \in X^*$ with $\|x^*\| < \varepsilon$ such that $(x^* -f) \colon X \to \mathbb{R} \cup \{-\infty\}$ has a strong maximum.