The relaxation of MaxCut can be formed as a vector program:
$$ \text{maximize} \quad \sum_{i,j}a_{ij}v_i^Tv_j \qquad \text{subject to} \quad u_i \in \mathbb{R}^n,\quad \|u_i\|_{L^2}=1 $$
although we know that this is equivalent to SDP which feasible region is spectrahedron, I am wondering the geometry of
$$\left\{ \mathbf{u}=(u_1,\cdots,u_n)^T, \text{where each }u_i\in\mathbb{R}^n, \|u_i\|_{L^2}=1 \right\}$$
What does it look like? is it convex?
A follow-up question is: does this two set equivalent?
$\{\mathbf{x}| \|x_i\|_{L_2}=1,\ \forall i\}$
$\{\mathbf{x}| \|x_i\|_{L_2}\leq 1,\ \forall i\}$
(a note: I added algebraic geometry tag here is because understanding high dimensional geometry such as spectrahedron is an direction in this subject.)