Pome47309C75E6C1Lemma 21.11.4.47309C75E6C1 Let $F\colon N\rightarrow [-\infty ,\infty ]$. Then $F$ is convex if and only if the following condition holds: suppose that $n,r\in N$ and $a,b\in \mathbb {R}$ such that $a>F(n)$, $b>F(r)$, then for any $t\in (0,1)$, we have
Proof. Assume first that the epigraph is convex. Given $a>F(n)$ and $b>F(r)$, choose finite $a'<a$ and $b'<b$ with $F(n)\leq a'$ and $F(r)\leq b'$. Convexity of the epigraph gives $F(tn+(1-t)r)\leq ta'+(1-t)b'<ta+(1-t)b$.
Conversely, take $(n,a),(r,b)$ in the epigraph. For every $\varepsilon>0$ the assumed inequality applied to $a+\varepsilon,b+\varepsilon$ gives $F(tn+(1-t)r)<ta+(1-t)b+\varepsilon$. Let $\varepsilon\downarrow0$. The resulting weak inequality says that the convex combination of these two points belongs to the epigraph. Endpoints cause no difficulty. This proof also covers values $-\infty$; points with value $+\infty$ contribute no points to the epigraph.
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