Generic Hahn–Banach results

Given f : X → R ∪ { + ∞ } a convex and lower semi-continuous function defined on a reflexive Banach space X, and L, a closed linear manifold of X over which f takes at least a real value, the aim of this note is to prove the following Baire category result: in the Euclidean setting, the set of affine functions dominated by f on L for which there is no dominated extension to X is always of first Baire category, but this set can be as large as a residual set, provided that X is a reflexive Banach space of infinite dimension.