Manuscript VII-E@1 · VII-E.tex · section · b713cf3c

A compact collar

One certified four-face slicing exchange › A compact collar · lines 2109–2135

VII-E@1#eq:cap, VII-E@1#eq:outer-rectangle, VII-E@1#eq:margins

Set $H = 2$ , and cap the graph by

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

Its non-resting 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} .

Indeed, $F_{s} \leq H + s$ is equivalent to the two displayed coordinate bounds. Fix the outer rectangle

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

Its collar margins are

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

Displayed formulas in this passage

VII-E@1#eq:cap · line 2112
$g_{s} (x, y) = min {F_{s} (x, y), H + s} .$
VII-E@1#eq:outer-rectangle · line 2124
$K = [- R_{x}, R_{x}] \times [- R_{y}, R_{y}], R_{x} = \frac{14}{3}, R_{y} = 6.$
VII-E@1#eq:margins · line 2130
$R_{x} - X_{s} = \frac{4 - s}{3} > 0, R_{y} - Y_{s} = \frac{4 - s}{5} > 0.$
unlabeled · line 2117
$R_{s} = [- X_{s}, X_{s}] \times [- Y_{s}, Y_{s}], X_{s} = \frac{10 + s}{3}, Y_{s} = \frac{26 + s}{5} .$