Proof. [03DZ]

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Proof.

Choose $\delta$ small enough to ensure that $\mathfrak{T}$ $+\ \delta(\tau_{0}-\widehat{\tau}_{0})\subset N$ for every simplex $\mathfrak{T}$ $=(\tau_{0}\prec\ldots\prec\tau_{r})\in K$ .

Figure 12: Choice of $\delta$ in the proof of Lemma 3.5.

Then the maximal cell which corresponds to $(\tau_{0},$ $\mathfrak{T}$ $)$ is given by

\delta\tau_{0}+(1-\delta)\text{\tiny$\mathfrak{T}$}\subseteq\text{\tiny$\mathfrak{T}$}+\delta(\tau_{0}-\widehat{\tau}_{0})\subset N.

∎