Affine minorant
= Affine minorant
{title2=$f(x)\geq\langle a,x\rangle+b$}
An affine minorant of $f$ is a function $x\mapsto\langle a,x\rangle+b$ that is everywhere at most $f(x)$. Every proper lower-semicontinuous <convex function> has such a minorant, by separating a point below its closed <epigraph>. Adding a positive quadratic then yields a <coercive function>, which establishes existence in proximal minimization.