Proposition 19.6.16.D70EFA3100C9 Let $(\Delta ^ i)_{i\in I}$ be an increasing net in $\mathcal{K}_ n$ with Hausdorff limit $\Delta $ such that $\operatorname{vol}\Delta ^ i>0$ for all $i\in I$. Consider an increasing net $(\Delta ^ i_{\bullet })_{i\in I}$ with $\Delta ^ i_{\bullet }\in \mathrm{TC}(\Delta ^ i)$ for all $i\in I$. Let $\Delta _{\max }=\lim _{i\in I}\Delta ^ i_{\max }$. For any $\tau <\Delta _{\max }$, let $\Delta _{\tau }$ be the Hausdorff limit of $\Delta ^ i_{\tau }$. Then $\Delta _{\bullet }\in \mathrm{TC}(\Delta )$ and
FDE24E7623A9
19.6.17
\begin{equation} \lim _{i\in I}G[\Delta ^ i_{\bullet }](a)=G[\Delta _{\bullet }](a) \end{equation}
for any $a\in \operatorname{Int}\Delta $.
Proof.
It is obvious that $\Delta _{\bullet }\in \mathrm{TC}(\Delta )$.
Fix $a\in \operatorname{Int}\Delta $. Since $\Delta ^ i\xrightarrow {d_{\mathrm{Haus}}}\Delta $, we have $a\in \Delta ^ i$ for all large enough $i$. By definition, the net $(G[\Delta ^ i_{\bullet }](a))_{i\in I}$ is increasing and the $\leq $ direction in (19.6.17) holds. Let us write $\tau =G[\Delta _{\bullet }](a)$ for the time being. By definition of $G$, for any $\epsilon >0$, we have
\[ a\in \Delta _{\tau -\epsilon /2}. \]
The concavity of $\Delta _{\bullet }$ guarantees that
\[ a\in \operatorname{Int}\Delta _{\tau -\epsilon }. \]
Indeed, choose $\sigma \ll 0$ so that a small ball around $a$ is contained in $\Delta _\sigma $. Since $a\in \Delta _{\tau -\epsilon /2}$ and
\[ \tau -\epsilon =t\sigma +(1-t)(\tau -\epsilon /2) \]
for a suitable $t\in (0,1)$, concavity gives a small ball around $a$ contained in $\Delta _{\tau -\epsilon }$.
Since $\Delta ^ i_{\tau -\epsilon }\xrightarrow {d_{\mathrm{Haus}}}\Delta _{\tau -\epsilon }$ and $a\in \operatorname{Int}\Delta _{\tau -\epsilon }$, it follows that for all large enough $i\in I$,
\[ a\in \Delta ^ i_{\tau -\epsilon }. \]
Therefore,
\[ \tau -\epsilon \leq G[\Delta ^ i_{\bullet }](a) \]
for all large enough $i$. Taking the limit with respect to $i$ and then with respect to $\epsilon $, we conclude the desired inequality.
$\square$
Pome
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