Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 218 / 1 / b / Solution

The one-variable constraint is ${\beta }^2\le t$, an interval whose endpoints expand as $\sqrt{t}$. The duplicated constraint is the disk ${\beta }_1^2+{\beta }_2^2\le t$; the loss contours are parallel strips perpendicular to $(1,1)$. Their first contact with the disk lies on ${\beta }_1={\beta }_2$. Equivalently, the fitted total coefficient is the least-squares coefficient clipped to $[-\sqrt{2t},\sqrt{2t}]$, compared with $[-\sqrt{t},\sqrt{t}]$ for one copy.