Is the following set a polyhedron?

Question

Is the following set a polyhedron, where $a_1, a_2 \in \mathbb{R}^{n}$? \begin{align} U = \{ X \in \mathbb{R}^{n \times n}: a^T_1Xa_1 \leq a^T_2 X a_2 \} \end{align}

Answer 1

Depends on what you mean by a polyhedron. $U$ is a linear halfspace orthogonal to the vector whose $i, j$-th coordinate is $v_{ij} = (a_1)_i (a_1)_j - (a_2)_i (a_2)_j.$