%file is s24.tex
Properties for bound $B_2$.
\begin{enumerate}
\item
$ \mu^* \leq B_2.$
\item
$f_2$ is convex, finite valued, with subdifferential
\[
\begin{array}{c}
\partial f_2(u) = \mbox{conv} \{ z=(1-y^2_i) \in \Re^n: \\ 
y=(y_i) \in \Re^n,~||y||^2=n,~f_2(u)=q_v(y)\}.
\end{array}
\]
\item
The bound $B_2$ is attained for some $u \in \Re^n$.
Moreover, if $B_2 > \mu^*$, then the hard case holds for $(RP^2_u).$
\end{enumerate}