A proper closed convex function has an affine minorant

Let $d$ be the dimension of the direction space of $\mathrm{aff}D$. If $d=0$, then $D$ is a singleton $\{x_0\}$ and the constant affine function $g(x_0)$ is a minorant. If $d>0$, choose affinely independent points $a_0,\dots ,a_d\in D$ spanning $\mathrm{aff}D$ and let $x_0$ be their barycenter. Iterating the two-point convexity condition shows their simplex lies in $D$. In affine barycentric coordinates, this simplex is defined by ${\lambda }_i\ge 0$ and ${\sum }_{i=0}^d{\lambda }_i=1$; at $x_0$ every ${\lambda }_i=1\mathrm{/}(d+1)$. These affine coordinate functions on the finite-dimensional space $\mathrm{aff}D$ are continuous, so they remain positive on a relative neighborhood of $x_0$, which therefore lies in the simplex and in $D$. Thus $x_0$ is a relative-interior point of $D$.

Put $p=(x_0,g(x_0))\in E_g$ and $q_j=(x_0,g(x_0)-1\mathrm{/}j)$ for integers $j\ge 1$. Each $q_j$ lies outside $E_g$ and tends to $p$. Let $p_j$ be its unique nearest point in $E_g$, supplied by [F1]. Since $p\in E_g$, $\mathrm{\parallel }{q}_j-p_j{\mathrm{\parallel }}_2\le 1\mathrm{/}j$. For each coordinate $\mathrm{\ell }<n+1$, $\mathrm{\mid }(q_j-p_j{)}_{\mathrm{\ell }}\mathrm{\mid }\le \mathrm{\parallel }{q}_j-p_j{\mathrm{\parallel }}_2\le 1\mathrm{/}j$, because each squared coordinate is a term of the sum defining the squared Euclidean norm. As $q_j\to p$ coordinatewise, this gives $p_j\to p$ coordinatewise and hence in Euclidean norm, since there are finitely many coordinates.

Normalize $u_j=(q_j-p_j)\mathrm{/}\mathrm{\parallel }{q}_j-p_j{\mathrm{\parallel }}_2$, which is defined because $q_j\mathrm{\notin }{E}_g$ and $E_g$ is closed. The vectors $u_j$ have norm one, so their range lies in the Euclidean ball $B(0,2)$. By [F3], choose a subsequence $(u_{j_{\mathrm{\ell }}})$ converging in ${\mathbb{R}}^{n+1}$ to $u=(a,c)$. Since each coordinate difference is bounded by the Euclidean norm, this convergence gives coordinate convergence; the finite-sum formula for the squared norm then gives $\mathrm{\parallel }u{\mathrm{\parallel }}_2^2={\lim }_{\mathrm{\ell }\to \mathrm{\infty }}\mathrm{\parallel }{u}_{j_{\mathrm{\ell }}}{\mathrm{\parallel }}_2^2=1$. Both $q_j$ and $p_j$ have horizontal coordinate in $\mathrm{aff}D$, so each horizontal component of $u_j$ lies in the direction space $V$ of $\mathrm{aff}D$. This finite-dimensional subspace is closed: after extending a basis of $V$ to one of ${\mathbb{R}}^n$, it is the common zero set of the remaining continuous coordinate functionals. Hence $a\in V$. For each fixed $z\in E_g$, [F2] gives $⟨u_j,z-p_j⟩\le 0$. Along the chosen subsequence, coordinate convergence and the finite-sum formula for the inner product show $⟨u_{j_{\mathrm{\ell }}},z-p_{j_{\mathrm{\ell }}}⟩\to ⟨u,z-p⟩$, so $⟨u,z-p⟩\le 0$.

The upward vertical ray $p+(0,s)$ lies in $E_g$ for all $s\ge 0$, so step 2.1 forces $c\le 0$. If $c=0$, the relative neighborhood from step 1.1 lets each direction $h\in V$ be tested at $x_0+\epsilon h$ and $x_0-\epsilon h$ for sufficiently small $\epsilon >0$. Applying step 2.1 to the corresponding epigraph points gives both $\epsilon ⟨a,h⟩\le 0$ and $-\epsilon ⟨a,h⟩\le 0$, so $⟨a,h⟩=0$ for every $h\in V$. Taking $h=a\in V$ gives $\mathrm{\parallel }a{\mathrm{\parallel }}_2^2=0$, hence $a=0$, contradicting $\mathrm{\parallel }u{\mathrm{\parallel }}_2=1$. Thus $c<0$.

For any $x\in D$, apply step 2.1 to $(x,g(x))\in E_g$ to get $⟨a,x-x_0⟩+c(g(x)-g(x_0))\le 0$. Dividing by $c<0$ reverses the inequality and yields $g(x)\ge g(x_0)+⟨-a\mathrm{/}c,x-x_0⟩$. Set $a_0=-a\mathrm{/}c$ and $b_0=g(x_0)-⟨a_0,x_0⟩$. This is the desired affine bound on $D$, and outside $D$ the value is $+\mathrm{\infty }$, so the bound holds on all of ${\mathbb{R}}^n$.