Solution
= Solution
For a positive semidefinite <matrix>, the sum of the <eigenvalues> is its trace. Thus
$$
S=\{M=M^T:M\succeq0,\ \operatorname{tr}M\leq s\}.
$$
If $M,N\in S$ and $0\leq t\leq1$, then $tM+(1-t)N\succeq0$ and
$$
\operatorname{tr}(tM+(1-t)N)
=t\operatorname{tr}M+(1-t)\operatorname{tr}N\leq s.
$$
\b[Hence $S$ is convex.]