Biconjugation as closed convexification ($f^{\ast \ast }=\mathrm{clconv}f$)

For a proper extended-real function with an affine minorant, biconjugate gives its greatest lower-semicontinuous convex minorant. Geometric separation of its epigraph turns containing closed half-spaces into affine lower bounds. Vertical half-spaces are recovered by adding multiples of a domain-separating inequality to one fixed affine minorant.