Formulation 1. is correct but 2. is not.

The RSOC constraint (t, u, v) \in RSOC translates to

t, u \geq 0, 2 \times t \times u \geq x_{1}^{2} + ... + x_{n}^{2}

Thus, to have w \times z \geq x^{2} + y^{2}, you can write:

`[w, z, x*sqrt(2), y*sqrt(2)] in RSOC()`

`[w/2, z, x, y] in RSOC()`

`[w, z/2, x, y] in RSOC()`

`[w/sqrt(2), z/sqrt(2), x, y] in RSOC()`

All these formulations are equivalent, i.e., they represent the same set.

To answer the second part your last post: this relation comes from the fact that a Rotated Second-Order Cone can be obtainedâ€¦ by rotating a second-order cone (see the Mosek modeling cookbook).

Specifically, the rotation matrix is

\begin{bmatrix}
\frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} \\
\frac{1}{\sqrt{2}} & -\frac{1}{\sqrt{2}} \\
&& 1\\
&&& \ddots\\
&&&& 1
\end{bmatrix}

Apply this change of coordinate to the SOC constraint \| 2x, 2y, w-z\| \leq (w+z)^{2} and youâ€™ll recover the original RSOC constraint.