Finite convex function as supremum of affine minorants ($\varphi (x)={\sup }_y[\varphi (y)+{\ell }_y(x-y)]$)

For a finite real convex function on an entire real vector space, apply the convex domination form of the Hahn-Banach theorem to $\varphi (y+z)-\varphi (y)$ and the zero functional on $\{0\}$. It supplies a linear functional ${\ell }_y$ dominated by this translated function. The affine function $a_y(x)=\varphi (y)+{\ell }_y(x-y)$ lies below $\varphi$ everywhere and equals it at $y$. Taking the supremum proves the formula. These are algebraic affine minorants; continuity requires additional topological hypotheses.