Consider the variance $($risk$)$ minimization problem $($Pr$1)$ in the notes with the definitions of

F$\_$a$,$ x$\_$min, R$\_$min, V$\_$min, R$\_$max.

$1.$ Prove that F$\_$a is a convex set.

$2.$ Assume that F$\_$a $!=.$ What are the conditions that guarantee that the feasible set of $($Pr$1)$ is not empty.

$3.$ Assume that F$\_$a $!=$ and that V $0.$

$($a$)$ Prove that the optimal value p$\_$P r$1$ is finite, and is attained, and the optimum is unique.

$($b$)$ Prove that the expected return target constraint is active at the optimum, i$.$e$.,$rT x$=$rp$.($Recall that an inequality constraint is active at a point if it holds with equality at that point.$)$