Affine minorant ($f(x)\ge ⟨a,x⟩+b$)

An affine minorant of a function $f$ on ${\mathbb{R}}^n$ is an affine function $x\mapsto a\cdot x+b$ satisfying $a\cdot x+b\le f(x)$ 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 $v\in {\mathbb{R}}^n$,The proximal objective is therefore coercive and lower semicontinuous; in finite dimensions it attains a minimum. Its strong convexity makes that minimizer unique, giving a well-defined proximal operator.^[ref] The finite-dimensional attainment argument should not be applied to arbitrary infinite-dimensional normed spaces merely from boundedness of a minimizing sequence.