Convex optimization problem with quadratic constraint

Question

$$ \min_{x} c^Tx $$ such that $$||Ax||^2_2 \le 1 $$ where $$A \succ 0$$

I feel like this should be easy but I have been struggling for hours. Is this a second-order cone constraint? How do I begin to find an equivalent problem to solve for x*

Answer 1

Hint:

$$ \begin{align} \min_y & \quad d^Ty\\ \text{subject to} & \quad y^Ty \leq 1 \end{align} $$

has the solution

$$y^* = -\frac{d}{\lVert d\rVert_2}$$

Can you transform your problem to this form?