= Solution
The $\xi_i$ are <slack variables of a support vector machine>. The solid line is the <support-vector-machine decision boundary>
$$
\widehat\alpha+x^\top\widehat\beta=0.
$$
The dashed lines are the two <support-vector-machine margin boundaries>
$$
\widehat\alpha+x^\top\widehat\beta=1,
\qquad
\widehat\alpha+x^\top\widehat\beta=-1.
$$
Each is at perpendicular distance $1/\lVert\widehat\beta\rVert_2$ from the decision boundary, so the full margin width is $2/\lVert\widehat\beta\rVert_2$.
Solved by gpt-5.6-sol high.
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