Cone reformulation

Question

Consider:

Where $v$ and $Y-X\beta$ are columns of the same length, say $n$. I would like to understand how to go from the first display to the second.

Answer 1

It is well known that the set $\{(x,s) : x^T x \leq st, s \geq 0\}$ can be represented as $$ \left\lVert \begin{pmatrix} 2x \\ t-s \end{pmatrix} \right\rVert_2 \leq t+s $$ Apply this with $x=Y-X\beta$, $s=v^T(Y-X\beta)$, and $t=t_0$.