Past exam of the mathematics course of the University of Cambridge / 2009 / iii / Paper 10 / 1 / d

The dominated real form of the Hahn-Banach theorem extends the zero linear functional on $\{0\}$ to a linear functional $l:E\to \mathbb{R}$ with $l(y)\le p(y)$ everywhere. If $x+z\in U$, convexity of the ball puts the whole segment in $U$. Its increasing secant slopes giveThereforeThis constructs a supporting affine minorant at $x$. Part (e) shows that it is a continuous supporting functional for a locally bounded convex function.