% file s21.tex
Properties for Bound $B_1$.
\begin{enumerate}
\item
$ \mu^* \leq B_1.  $
\item
$f_1$ is convex, finite valued, with subdifferential
\[
\begin{array}{c}
  \partial f_1(v) = \conv \{ z=(1-x^2_i) \in \Re^n:\\
    x=(x_i) \in \Re^n 
     -1 \leq x \leq 1,~f_1(v)=q_v(x)\}.
\end{array}
\]
\item
$B_1$ is attained for some $v \in \Re^n$ such that
$\lambda_{max}(Q - \diag(v))=0.$
\item
\[B_1 = \inf_{Q-\diag(v) < 0} f_1(v).
\]
\end{enumerate}