= Solution
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 $l(y)\leq p(y)$ everywhere. If $x+z\in U$, <convexity> of the ball puts the whole segment in $U$. Its increasing secant slopes give
$$
p(z)=\lim_{t\downarrow0}\frac{f(x+tz)-f(x)}t\leq f(x+z)-f(x).
$$
Therefore
$$
\boxed{f(x+z)\geq f(x)+l(z)\qquad(x+z\in U).}
$$
This constructs a supporting <affine minorant> at $x$. Part (e) shows that it is a <continuous supporting functional for a locally bounded convex function>.
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