# linear functions/hyperplanes vs. convex functions/convex sets in Hilbert space

The simplest Hahn-Banach extension theorem in Hilbert space $$X$$ avoids the use of the axiom of choice by virtue of the Riesz representation theorem. But what about the version of the theorem where the sought-for linear functional is required to remain below a given sub-linear or convex function? Also, to which extent can we separate 2 disjoint convex sets by a hyperplane without Zorn? Can we assert that any hyperplane $$H\subset X$$ has a translate that is tangent (*) to a given bounded closed convex subset $$C\subset X$$?

This question has an identical counterpart on stackexchange.

(*) "tangent" means that the linear bounded function whose kernel represents the hyperplane is non-negative on $$C$$ but not strictly positive.

• To separate a point $x$ from a closed convex set $C$, one can take the best approximation $y\in C$ to $x$ and a hyperplane perpendicular to $x-y$. – Dirk Werner Feb 27 at 15:32
• And similarly one can find nearest points between closed convex set and a disjoint compact convex body, but that's about "the best" that I'm aware of. – Thibaut Demaerel Feb 27 at 17:01