A few questions related to SOCP:
[1] Similar to this question, is there any geometric / intuitive way of seeing the following (rotated cone) relationship: \begin{align} \left\| \begin{bmatrix} Ax \\ (1 + b^Tx + c)/2 \end{bmatrix} \right\|_2 \leq (1 - b^Tx - c)/2 \iff x^T A^T A x + b^Tx + c \leq 0 \tag{1} \end{align}
I get the algebra on the left of the "$\iff$" to work out by considering the squares of each side, but I don't have a sense for how we intuitively formulate this in the first place. Based on @MichaelGrant's comment, I now see that squaring both sides works in this case because both are nonnegative (and in general, this should hold, but note that it may not yield a CVX-representable constraint). So, I'm trying to understand it in the context of thinking about inner products (which leads to thinking about it in terms of SDPs below).
In particular, we can represent $Q_n = \{ (x,t): \|x\|_2 \leq t,\, t \geq 0,\, x \in \mathbb{R}^n \}$ as \begin{align} Q_n = \left\{ (x,t): [x^T, t] \begin{bmatrix}I & \\ & -1 \end{bmatrix} \begin{bmatrix} x \\ t \end{bmatrix}\leq 0,\, t \geq 0 \right\} \tag{2} \end{align} but I haven't been able to represent $x^T A^T A x + b^T x+ c \leq 0$ in this way -- how can we do so? Do we have off-diagonal terms for rotated SO cones?
[2] I see how we can use the Schur complement to reframe a SOCP as a SDP -- for example, if we consider $$ \|Ax + b\|_2 \leq c^T x + d, \tag{3} $$ I see how the Schur complement of \begin{align} \begin{bmatrix} I(c^T x + d) & Ax + b \\ (Ax + b)^T & c^Tx + d \end{bmatrix} \succeq 0 \tag{4} \end{align} gives constraint (3). However, I had trouble representing the SOCP constraint with the inner-product formulation from (2) -- how do we do this? I tried $$ [x^T A^T,\, c^T x,\, -b,\, -d] \begin{bmatrix} I & & &\\ & 1 & & \\ & & -I & \\ & & & -1 \end{bmatrix} [x^T A^T,\, c^T x,\, -b,\, -d]^T \tag{5} $$ but was getting stuck.
[3] Finally, how should I think about going back and forth between an inner product formulation like (2), a standard SOCP constraint like (3), and a SDP like (4) -- i.e., what is the context for each formulation?