 Semidefinite relaxation of slab-constrained quadratic maximization

Replacing $x{x}^T$ by a general positive semidefinite matrix relaxes maximization of $\Vert x{\Vert }_2^2$ subject to $|a_i^{T}x|\le 1$. The relaxed value is an upper bound because $x{x}^T$ has trace $\Vert x{\Vert }_2^2$ and satisfies the same quadratic constraints. Both problems are unbounded if the $a_i$ fail to span the ambient space. If they span it, $H={\sum }_i{a}_i{a}_i^T$ is positive definite and ${\lambda }_{\min }(H)\mathrm{tr}X\le \mathrm{tr}(HX)\le m$, proving boundedness and attainment of the relaxation.