Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 205 / 1 / b / Solution

Only the $k$th block can have nonzero subgradient coordinates. If ${\beta }^{(k)}\ne 0$, the Cauchy-Schwarz inequality shows that the unique supporting vector is ${\beta }^{(k)}/\Vert {\beta }^{(k)}{\Vert }_2$. At zero, the defining inequality is $\Vert h{\Vert }_2\ge u^{T}h$ for every block vector $h$, which is equivalent to $\Vert u{\Vert }_2\le 1$. Thus