I have come across two different definitions of the second-order/Lorentz cone. The first is the standard form where $t$ is a scalar and $\mathbf{y} \in \mathbb{R}^n$.
$$
\mathcal{C}_1 = \bigg\{ \begin{bmatrix}\mathbf{y} \\ t\end{bmatrix} \in \mathbb{R}^{n+1}: t \ge \lVert \mathbf{y}\rVert_2\bigg\}
$$
The second definition is from this paper. Here the cone is defined for $\mathbf{x} \in \mathbb{R}^{n+1}$.
$$
\mathcal{C}_2 = \big\{\mathbf{x} : \mathbf{x}^{\sf T} \mathbf{1} \ge \sqrt{n}\lVert \mathbf{x}\rVert_2 \big\}
$$
Are $\mathcal{C}_1$ and $\mathcal{C}_2$ the same?
Best Answer
The axis of the first cone is in the direction $(0,0,\ldots,0,1)$, while the axis of the second cone is in the direction $(1, 1, \ldots, 1)$. There is also some scaling difference in the sharpness of the cone (the factor $\sqrt{n}$).