Lemma 21.4.6.F7AECD623994 Let $m\geq 1$ and $\beta _1,\ldots ,\beta _ m,\alpha \in M_{\mathbb R}$. The function
\[ x\longmapsto \log \sum _{i=1}^ m\mathrm e^{\langle \beta _ i,x\rangle } -\langle \alpha ,x\rangle \]
is bounded below on $N_{\mathbb R}$ if and only if $\alpha $ belongs to the convex hull of $\beta _1,\ldots ,\beta _ m$. Suppose that all $\beta _ i$ and $\alpha $ lie in $M$, identify $M$ with $\mathbb Z^ n$ by a basis and $N_{\mathbb R}$ with $\mathbb R^ n$ by the dual basis, and write $z^\alpha =z_1^{\alpha _1}\cdots z_ n^{\alpha _ n}$. Then this condition is also equivalent to the boundedness on $(\mathbb C^*)^ n$ of
\[ \frac{|z^\alpha |^2}{\sum _ i|z^{\beta _ i}|^2}. \]
Proof.
If $\alpha =\sum _ i\lambda _ i\beta _ i$ is a convex combination, then $\langle \alpha ,x\rangle \leq \max _ i\langle \beta _ i,x\rangle \leq \log \sum _ i\mathrm e^{\langle \beta _ i,x\rangle }$, so the function is nonnegative. Conversely, suppose that $\alpha $ is not in the convex hull. The hull is compact, being the image of the standard simplex under the linear map $(\lambda _ i)\mapsto \sum _ i\lambda _ i\beta _ i$, and convex, so lemma 21.2.3(1), applied in $M_{\mathbb R}$, gives $v\in N_{\mathbb R}$ with $\langle \alpha ,v\rangle >\max _ i\langle \beta _ i,v\rangle $. Since the logarithmic sum differs from $\max _ i\langle \beta _ i,x\rangle $ by a number in $[0,\log m]$, the value of the function at $x=tv$ is at most $t(\max _ i\langle \beta _ i,v\rangle -\langle \alpha ,v\rangle )+\log m$, which tends to $-\infty $ as $t\to +\infty $. For the last assertion, put $x_ j=\log |z_ j|^2$. Then $|z^\gamma |^2=\mathrm e^{\langle \gamma ,x\rangle }$ for every $\gamma \in \mathbb Z^ n$, including exponents with negative entries, so the displayed quotient equals the exponential of minus the function above; and $z\mapsto x$ maps $(\mathbb C^*)^ n$ onto $\mathbb R^ n$.
$\square$
Pome
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