I am taking my first course in functional analysis with applications to optimization. I am confused about why there is not another Hahn-Banach theorem that applies to convex/concave functionals. To specify what I am confused about, I provide the definitions that I am using and walk you through my thought process and intuition. My question is at the bottom of this post in bold.
I am thinking specifically about the extension form of the theorem, as opposed to Mazur's geometric version of the theorem. In particular, consider the following statement of the Hahn-Banach Theorem:
Let $X$ be a real linear normed space and $p$ a continuous sublinear functional on $X$. Let $f$ be a linear functional defined on a subspace $M$ of $X$ satisfying $f(m) \leq p(m)$ for all $m \in M$. Then there is an extension $F$ of $f$ from $M$ to $X$ such that $F(x) \leq p(x)$ on $X$.
I learned about concave functionals in the context of Fenchel's duality theorem. They were defined as follows: a real-valued functional $f$ defined on a convex subset $C$ of a linear vector space is concave if $\alpha f(x_1) + (1-\alpha) f(x_2) \leq f(\alpha x_1 + (1-\alpha) x_2)$, i.e. the value at the interpolated input exceeds the value of the interpolation of the outputs.
Moreover, I am using the definition of a sublinear functional as a real-valued function $p$ defined on a real vector space $X$ such that $p(x_1+x_2)\leq p(x_1)+p(x_2)$ and $p(\alpha x) = \alpha p(x)$.
In optimization, the Hahn-Banach theorem guarantees the existence of a hyperplane that is tangent to a convex set, allowing us to optimize, provided that information about constraints is encoded in the convex set that we are optimizing over. Also, in Fenchel duality, the same ideas come into play. Although Fenchel duality is a more complicated approach, it allows us to use constraints that are not necessarily linear and find global extrema (in a convex set specified our constraints). The Fenchel duality theorem requires that we optimize over an intersection of convex sets (which is necessarily convex and even in the infinite-dimensional case by Zorn's lemma). These considerations motivate the following question:
Why is there no analog to the Hahn-Banach theorem for concave/convex functionals that extends the convex sets in the conjugate space specified by conjugate functionals?
Thanks in advance!