= Affine minorant
{title2=$f(x)\geq\langle a,x\rangle+b$}
An affine minorant of a <function> $f$ on $\mathbb R^n$ is an <affine function> $x\mapsto a\cdot x+b$ satisfying $a\cdot x+b\leq f(x)$ everywhere. Every <proper convex function> that is <lower semicontinuous> on $\mathbb R^n$ has an affine minorant: separate a point strictly below its <closed> <epigraph> by a <hyperplane>. The finiteness of $f$ somewhere makes the separating vertical coefficient strictly positive, so the separator can be solved for height.
This supplies a useful existence estimate. For $\tau>0$ and $v\in\mathbb R^n$,
$$
f(x)+\frac{\|x-v\|^2}{2\tau}\geq a\cdot x+b+\frac{\|x-v\|^2}{2\tau}\longrightarrow+\infty\qquad(\|x\|\to\infty).
$$
The proximal objective is therefore <coercive> and <lower semicontinuous>; in finite dimensions it attains a minimum. Its <strongly convex function>[strong convexity] makes that minimizer unique, giving a well-defined <proximal operator>.<Proximal Algorithms by Neal Parikh and Stephen Boyd>[§1.1, printed pp124–125, existence and uniqueness for proper closed convex functions]{ref} The finite-dimensional attainment argument should not be applied to arbitrary infinite-dimensional <normed spaces> merely from boundedness of a minimizing sequence.
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