Definition 18.3.1 (Affine Minorant).label Let $\dpn{E, F}{\lambda}$ be a duality over $\real$, $f: E \to (-\infty, \infty]$, and $(\phi, \alpha) \in F \times \real$, then the pair $(\phi, \alpha)$ is an affine minorant of $f$, denoted $(\phi, \alpha) \le f$, if
\[\dpn{x, \phi}{\lambda}- \alpha \le f(x) \quad \forall x \in E\]
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