Lemma 9.11.195D5A847F99AC

Proof. Set $F=g-\ell $. Suppose its recession is positive on $C\setminus \{ 0\} $. Then $F(x)\geq \gamma |x|-A$ on $C$ for some $\gamma >0$ and $A$. Otherwise there are $x_ j\in C$ with $|x_ j|\to \infty $ and $F(x_ j)/|x_ j|\leq 1/j$. After taking a subsequence, $x_ j/|x_ j|\to v\in C$ with $|v|=1$. For fixed $t>0$, convexity gives

Continuity gives $F(tv)\leq F(0)$ for every $t$, contradicting positive recession. A bound on the remaining compact set gives the asserted $A$. Exponential decay proves integrability. The same bound persists when $\ell $ is changed by a sufficiently small linear form, so $\ell $ lies in the stated interior.

Conversely, if $(\operatorname {rec}F)(v)\leq 0$ for some nonzero $v\in C$, then $F(tv)\leq F(0)$ for $t\geq 0$. Choose a compact ball $K$ in the interior of $C$. On the half-tube $\{ tv+x/2:t\geq 1,\ x\in K\} $, convexity gives

Thus $\mathrm e^{-2F}$ has a positive lower bound on a set of infinite volume. If instead $\ell $ is in the stated interior, for each nonzero $v\in C$ one may perturb $\ell $ within that interior by a linear form positive on $v$. Dividing its defining upper bound along $tv$ by $t$ gives the strict recession inequality for $\ell $. Finally, translating $C$ replaces $g(x)$ by $g(x+b)$ and multiplies the exponential by a constant; recession is unchanged. $\square$