Manuscript En03-A@1 · En03-A.tex · section · 5398614e

Compact collar and frozen exterior

External conical geometry and certified local moves › Compact collar and frozen exterior · lines 1718–1745

En03-A@1#eq:capped, En03-A@1#eq:inner-rectangle, En03-A@1#eq:K, En03-A@1#eq:collar-margins

For $- 1 \leq s \leq 1$ and $H = 2$ , define

g_{s} (x, y) = min {F_{s} (x, y), H + s} .

The uncapped interior region is

R_{s} = [- X_{s}, X_{s}] \times [- Y_{s}, Y_{s}], X_{s} = \frac{10 + s}{3}, Y_{s} = \frac{26 + s}{5},

since $F_{s} \leq H + s$ is equivalent to these two inequalities. Choose

K = [- R_{x}, R_{x}] \times [- R_{y}, R_{y}], R_{x} = \frac{14}{3}, R_{y} = 6.

Then

R_{x} - X_{s} = \frac{4 - s}{3} > 0, R_{y} - Y_{s} = \frac{4 - s}{5} > 0.

Displayed formulas in this passage

En03-A@1#eq:capped · line 1721
$g_{s} (x, y) = min {F_{s} (x, y), H + s} .$
En03-A@1#eq:inner-rectangle · line 1726
$R_{s} = [- X_{s}, X_{s}] \times [- Y_{s}, Y_{s}], X_{s} = \frac{10 + s}{3}, Y_{s} = \frac{26 + s}{5},$
En03-A@1#eq:K · line 1733
$K = [- R_{x}, R_{x}] \times [- R_{y}, R_{y}], R_{x} = \frac{14}{3}, R_{y} = 6.$
En03-A@1#eq:collar-margins · line 1739
$R_{x} - X_{s} = \frac{4 - s}{3} > 0, R_{y} - Y_{s} = \frac{4 - s}{5} > 0.$