Proposition 19.4.39.63CE446DCB89 As $\mathbb {Q}_{>0}\ni \delta \searrow 0$, the convex bodies $\Delta _{\nu }(\theta +\delta \omega +\mathrm{dd}^{\mathrm{c}}\varphi )$ are decreasing and
\[ \Delta _{\nu }(\theta +\delta \omega +\mathrm{dd}^{\mathrm{c}}\varphi )\xrightarrow {d_{\mathrm{Haus}}} \Delta _{\nu }(\theta _{\varphi }). \]
Proof.
Let $0\leq \delta <\delta '$ be two rational numbers. Take $C\in \mathbb {N}_{>0}$ divisible enough, so that $C\delta $ and $C\delta '$ are both integers. Then by proposition 19.4.33,
\[ \Delta _{\nu }(C\theta +C\delta \omega +C\mathrm{dd}^{\mathrm{c}}\varphi )\subseteq \Delta _{\nu }(C\theta +C\delta '\omega +C\mathrm{dd}^{\mathrm{c}}\varphi ). \]
It follows that
\[ \Delta _{\nu }(\theta +\delta \omega +\mathrm{dd}^{\mathrm{c}}\varphi )\subseteq \Delta _{\nu }(\theta +\delta '\omega +\mathrm{dd}^{\mathrm{c}}\varphi ). \]
On the other hand,
\[ \operatorname{vol}\Delta _{\nu }(\theta +\delta \omega +\mathrm{dd}^{\mathrm{c}}\varphi )=\frac{1}{n!}\operatorname{vol}(\theta +\delta \omega )_{\varphi }. \]
As $\delta \to 0+$, the right-hand side converges to
\[ \operatorname{vol}\Delta _{\nu }(\theta ,\varphi )=\frac{1}{n!}\operatorname{vol}\theta _{\varphi }, \]
thanks to proposition 18.1.27. Our assertion therefore follows.
$\square$
Pome
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