Reference on vector-valued convex conjugate

Question

The following definition of convex conjugate is taken from Wiki:

Let $X$ be a real topological vector space, and let $X^*$ be the dual space to $X.$ Denote the dual pairing by $$\langle \cdot ,\cdot \rangle :X^{*}\times X\to \mathbb {R}.$$ For a function $f:X\to\mathbb{R}\cup \{\pm\infty\}$taking values on the extended real number line, the convex conjugate $$f^{*}:X^{*}\to \mathbb {R} \cup \{-\infty ,+\infty \}$$ is defined in terms of the supremum by $$f^{*}\left(x^{*}\right):=\sup \left\{\left.\left\langle x^{*},x\right\rangle -f\left(x\right)\right|x\in X\right\}.$$

Bachir introduced the notion conjugate $f^\times$ of $f:X\to\mathbb{R}$ as $$f^\times (\phi) := \sup_{x\in X}\{\phi(x) - f(x)\}$$ for all $\phi\in C_b(X),$ the set of all bounded real-valued continuous functions on $X.$

Question: Do we have convex conjugate for vector-valued function? More precisely, if $E$ is a Banach space, can we define convex conjugate of $f:X\to E$ by $$\tilde{f}^\times(\phi) = \sup_{x\in X}\{\|\phi(x)\| - \|f(x)\|\}$$ for all $\phi\in C_b(X,E),$ the set of all bounded $E$-valued continuous functions on $X?$

Answer 1

I haven't seen this notion. Also note that Bachir did not use the whole space of continuous bounded functions for $\phi$ but certain subsets (doesn't the biconjugate get weird if you use the full space?).

Two pointers:

• There are various notions of abstract convexity and some of them do feature a generalized notion of conjugation (I would start to look as Singer's "Abstract Convex Analysis").

• There is the notion of $c$-convexity, and $c$-conjugation used in optimal transport (see the books "Optimal Transport for Applied Mathematicians" by Santambrogio or "Topics in Optimal Transportation" by Villani, for example). This does not generalize to vector valued functions, but may still be worth looking at.