Manuscript En06-A@1 · En06-A.tex · theorem · e04c9892

Theorem: Uniform two-parameter shape theorem

Increasing stencils: the geometric escape · lines 1069–1091 · proof included or linked

En06-A@1#thm:shape, En06-A@1#eq:shapepoly, En06-A@1#eq:shape

For every $n$ , $k \geq 0$ , and $x \in Λ_{n}$ ,

d_{H} (h_{n} B_{S_{n}} (k), k ε_{n} P_{n}) d_{H} (B_{d_{n}} (x, k), B_{2} (x, k ε_{n})) \leq (D + 1) ε_{n}, \leq (D + 1) ε_{n} + 2 δ_{n} k ε_{n} .

In particular, the second estimate tends uniformly to zero for $k ε_{n} \leq R < \infty$ .

Proof. Apply Lemma lem:filling with maximum generator length at most $L_{n}$ , then multiply by $h_{n}$ . This gives (eq:shapepoly). Translation invariance and Lemma lem:polyapprox give

d_{H} (k ε_{n} P_{n}, k ε_{n} B_{2}) \leq 2 δ_{n} k ε_{n} .

Use the triangle inequality.

Displayed formulas in this passage

En06-A@1#eq:shapepoly · line 1071
$d_{H} (h_{n} B_{S_{n}} (k), k ε_{n} P_{n}) d_{H} (B_{d_{n}} (x, k), B_{2} (x, k ε_{n})) \leq (D + 1) ε_{n}, \leq (D + 1) ε_{n} + 2 δ_{n} k ε_{n} .$
En06-A@1#eq:shape · line 1071
$d_{H} (h_{n} B_{S_{n}} (k), k ε_{n} P_{n}) d_{H} (B_{d_{n}} (x, k), B_{2} (x, k ε_{n})) \leq (D + 1) ε_{n}, \leq (D + 1) ε_{n} + 2 δ_{n} k ε_{n} .$
unlabeled · line 1087
$d_{H} (k ε_{n} P_{n}, k ε_{n} B_{2}) \leq 2 δ_{n} k ε_{n} .$

Theorem: Uniform two-parameter shape theorem

Displayed formulas in this passage

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