Proposition 3.2 . [00Q4]

Original official author HTML, exact retained edition. Historical TeX conversion verdicts remain unchanged. Cited-edition alignment and mathematical self-containment are not assessed.

Complete original source context · Original author HTML

Proposition 3.2.

(cf. [23, Prop. 3.2]) The amoebas $\mathcal{A}_{\lambda}^{s}$ converge in the Hausdorff distance to $\mathcal{A}_{\lambda}^{\infty}$ in the Hausdorff distance as $s\to\infty$ . In fact

\begin{cases}\text{dist}_{\mathbb{R}^{n+1}}(x,\mathcal{A}_{\lambda}^{\infty})\leq\frac{C}{s},\quad\forall x\in\text{Log}_{s}(X_{s}),\\ \text{dist}_{\mathbb{R}^{n+1}}(x,\text{Log}_{s}(X_{s}))\leq\frac{C}{s},\quad\forall x\in\mathcal{A}_{\lambda}^{\infty}.\end{cases}